Algebra Examples

$\frac{1}{x (5 x + 3) (5 x^{2} + 3)}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $B$ .

$\frac{A}{x} + \frac{B}{5 x + 3}$

Step 1.2

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor is 2nd order, $2$ terms are required in the numerator. The number of terms required in the numerator is always equal to the order of the factor in the denominator.

$\frac{A}{x} + \frac{B}{5 x + 3} + \frac{C x + D}{5 x^{2} + 3}$

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $x (5 x + 3) (5 x^{2} + 3)$ .

$\frac{1 (x (5 x + 3) (5 x^{2} + 3))}{x (5 x + 3) (5 x^{2} + 3)} = \frac{A (x (5 x + 3) (5 x^{2} + 3))}{x} + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.4

Cancel the common factor of $x$ .

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Step 1.4.1

Cancel the common factor.

Step 1.4.2

Rewrite the expression.

$\frac{1 ((5 x + 3) (5 x^{2} + 3))}{(5 x + 3) (5 x^{2} + 3)} = \frac{A (x (5 x + 3) (5 x^{2} + 3))}{x} + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.5

Cancel the common factor of $5 x + 3$ .

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Step 1.5.1

Cancel the common factor.

Step 1.5.2

Rewrite the expression.

$\frac{1 (5 x^{2} + 3)}{5 x^{2} + 3} = \frac{A (x (5 x + 3) (5 x^{2} + 3))}{x} + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.6

Cancel the common factor of $5 x^{2} + 3$ .

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Step 1.6.1

Cancel the common factor.

$\frac{1 (5 x^{2} + 3)}{5 x^{2} + 3} = \frac{A (x (5 x + 3) (5 x^{2} + 3))}{x} + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.6.2

Rewrite the expression.

$1 = \frac{A (x (5 x + 3) (5 x^{2} + 3))}{x} + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7

Simplify each term.

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Step 1.7.1

Cancel the common factor of $x$ .

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Step 1.7.1.1

Cancel the common factor.

$1 = \frac{A (x (5 x + 3) (5 x^{2} + 3))}{x} + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.1.2

Divide $A ((5 x + 3) (5 x^{2} + 3))$ by $1$ .

$1 = A ((5 x + 3) (5 x^{2} + 3)) + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.2

Expand $(5 x + 3) (5 x^{2} + 3)$ using the FOIL Method.

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Step 1.7.2.1

Apply the distributive property.

$1 = A (5 x (5 x^{2} + 3) + 3 (5 x^{2} + 3)) + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.2.2

Apply the distributive property.

$1 = A (5 x (5 x^{2}) + 5 x \cdot 3 + 3 (5 x^{2} + 3)) + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.2.3

Apply the distributive property.

$1 = A (5 x (5 x^{2}) + 5 x \cdot 3 + 3 (5 x^{2}) + 3 \cdot 3) + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.3

Simplify each term.

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Step 1.7.3.1

Rewrite using the commutative property of multiplication.

$1 = A (5 \cdot (5 x \cdot x^{2}) + 5 x \cdot 3 + 3 (5 x^{2}) + 3 \cdot 3) + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.3.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.7.3.2.1

Move $x^{2}$ .

$1 = A (5 \cdot (5 (x^{2} x)) + 5 x \cdot 3 + 3 (5 x^{2}) + 3 \cdot 3) + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.3.2.2

Multiply $x^{2}$ by $x$ .

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Step 1.7.3.2.2.1

Raise $x$ to the power of $1$ .

$1 = A (5 \cdot (5 (x^{2} x)) + 5 x \cdot 3 + 3 (5 x^{2}) + 3 \cdot 3) + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.3.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 = A (5 \cdot (5 x^{2 + 1}) + 5 x \cdot 3 + 3 (5 x^{2}) + 3 \cdot 3) + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.3.2.3

Add $2$ and $1$ .

$1 = A (5 \cdot (5 x^{3}) + 5 x \cdot 3 + 3 (5 x^{2}) + 3 \cdot 3) + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.3.3

Multiply $5$ by $5$ .

$1 = A (25 x^{3} + 5 x \cdot 3 + 3 (5 x^{2}) + 3 \cdot 3) + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.3.4

Multiply $3$ by $5$ .

$1 = A (25 x^{3} + 15 x + 3 (5 x^{2}) + 3 \cdot 3) + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.3.5

Multiply $5$ by $3$ .

$1 = A (25 x^{3} + 15 x + 15 x^{2} + 3 \cdot 3) + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.3.6

Multiply $3$ by $3$ .

$1 = A (25 x^{3} + 15 x + 15 x^{2} + 9) + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.4

Apply the distributive property.

$1 = A (25 x^{3}) + A (15 x) + A (15 x^{2}) + A \cdot 9 + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.5

Simplify.

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Step 1.7.5.1

Rewrite using the commutative property of multiplication.

$1 = 25 A x^{3} + A (15 x) + A (15 x^{2}) + A \cdot 9 + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.5.2

Rewrite using the commutative property of multiplication.

$1 = 25 A x^{3} + 15 A x + A (15 x^{2}) + A \cdot 9 + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.5.3

Rewrite using the commutative property of multiplication.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + A \cdot 9 + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.5.4

Move $9$ to the left of $A$ .

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 \cdot A + \frac{(B) (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.6

Cancel the common factor of $5 x + 3$ .

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Step 1.7.6.1

Cancel the common factor.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + \frac{B (x (5 x + 3) (5 x^{2} + 3))}{5 x + 3} + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.6.2

Divide $(B) (x (5 x^{2} + 3))$ by $1$ .

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + (B) (x (5 x^{2} + 3)) + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.7

Apply the distributive property.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + B (x (5 x^{2}) + x \cdot 3) + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.8

Rewrite using the commutative property of multiplication.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + B (5 x \cdot x^{2} + x \cdot 3) + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.9

Move $3$ to the left of $x$ .

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + B (5 x \cdot x^{2} + 3 \cdot x) + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.10

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.7.10.1

Move $x^{2}$ .

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + B (5 (x^{2} x) + 3 \cdot x) + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.10.2

Multiply $x^{2}$ by $x$ .

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Step 1.7.10.2.1

Raise $x$ to the power of $1$ .

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + B (5 (x^{2} x) + 3 \cdot x) + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + B (5 x^{2 + 1} + 3 \cdot x) + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.10.3

Add $2$ and $1$ .

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + B (5 x^{3} + 3 \cdot x) + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + B (5 x^{3} + 3 x) + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.11

Apply the distributive property.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + B (5 x^{3}) + B (3 x) + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.12

Rewrite using the commutative property of multiplication.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + B (3 x) + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.13

Rewrite using the commutative property of multiplication.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.14

Cancel the common factor of $5 x^{2} + 3$ .

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Step 1.7.14.1

Cancel the common factor.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + \frac{(C x + D) (x (5 x + 3) (5 x^{2} + 3))}{5 x^{2} + 3}$

Step 1.7.14.2

Divide $(C x + D) (x (5 x + 3))$ by $1$ .

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + (C x + D) (x (5 x + 3))$

Step 1.7.15

Apply the distributive property.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + (C x + D) (x (5 x) + x \cdot 3)$

Step 1.7.16

Rewrite using the commutative property of multiplication.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + (C x + D) (5 x \cdot x + x \cdot 3)$

Step 1.7.17

Move $3$ to the left of $x$ .

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + (C x + D) (5 x \cdot x + 3 \cdot x)$

Step 1.7.18

Multiply $x$ by $x$ by adding the exponents.

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Step 1.7.18.1

Move $x$ .

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + (C x + D) (5 (x \cdot x) + 3 \cdot x)$

Step 1.7.18.2

Multiply $x$ by $x$ .

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + (C x + D) (5 x^{2} + 3 \cdot x)$

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + (C x + D) (5 x^{2} + 3 x)$

Step 1.7.19

Expand $(C x + D) (5 x^{2} + 3 x)$ using the FOIL Method.

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Step 1.7.19.1

Apply the distributive property.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + C x (5 x^{2} + 3 x) + D (5 x^{2} + 3 x)$

Step 1.7.19.2

Apply the distributive property.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + C x (5 x^{2}) + C x (3 x) + D (5 x^{2} + 3 x)$

Step 1.7.19.3

Apply the distributive property.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + C x (5 x^{2}) + C x (3 x) + D (5 x^{2}) + D (3 x)$

Step 1.7.20

Simplify each term.

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Step 1.7.20.1

Rewrite using the commutative property of multiplication.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + 5 C x \cdot x^{2} + C x (3 x) + D (5 x^{2}) + D (3 x)$

Step 1.7.20.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.7.20.2.1

Move $x^{2}$ .

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + 5 C (x^{2} x) + C x (3 x) + D (5 x^{2}) + D (3 x)$

Step 1.7.20.2.2

Multiply $x^{2}$ by $x$ .

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Step 1.7.20.2.2.1

Raise $x$ to the power of $1$ .

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + 5 C (x^{2} x) + C x (3 x) + D (5 x^{2}) + D (3 x)$

Step 1.7.20.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + 5 C x^{2 + 1} + C x (3 x) + D (5 x^{2}) + D (3 x)$

Step 1.7.20.2.3

Add $2$ and $1$ .

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + 5 C x^{3} + C x (3 x) + D (5 x^{2}) + D (3 x)$

Step 1.7.20.3

Rewrite using the commutative property of multiplication.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + 5 C x^{3} + 3 C x \cdot x + D (5 x^{2}) + D (3 x)$

Step 1.7.20.4

Multiply $x$ by $x$ by adding the exponents.

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Step 1.7.20.4.1

Move $x$ .

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + 5 C x^{3} + 3 C (x \cdot x) + D (5 x^{2}) + D (3 x)$

Step 1.7.20.4.2

Multiply $x$ by $x$ .

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + 5 C x^{3} + 3 C x^{2} + D (5 x^{2}) + D (3 x)$

Step 1.7.20.5

Rewrite using the commutative property of multiplication.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + 5 C x^{3} + 3 C x^{2} + 5 D x^{2} + D (3 x)$

Step 1.7.20.6

Rewrite using the commutative property of multiplication.

$1 = 25 A x^{3} + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + 5 C x^{3} + 3 C x^{2} + 5 D x^{2} + 3 D x$

Step 1.8

Reorder.

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Step 1.8.1

Move $A$ .

$1 = 25 x^{3} A + 15 A x + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + 5 C x^{3} + 3 C x^{2} + 5 D x^{2} + 3 D x$

Step 1.8.2

Move $A$ .

$1 = 25 x^{3} A + 15 x A + 15 A x^{2} + 9 A + 5 B x^{3} + 3 B x + 5 C x^{3} + 3 C x^{2} + 5 D x^{2} + 3 D x$

Step 1.8.3

Move $A$ .

$1 = 25 x^{3} A + 15 x A + 15 x^{2} A + 9 A + 5 B x^{3} + 3 B x + 5 C x^{3} + 3 C x^{2} + 5 D x^{2} + 3 D x$

Step 1.8.4

Move $B$ .

$1 = 25 x^{3} A + 15 x A + 15 x^{2} A + 9 A + 5 x^{3} B + 3 B x + 5 C x^{3} + 3 C x^{2} + 5 D x^{2} + 3 D x$

Step 1.8.5

Move $B$ .

$1 = 25 x^{3} A + 15 x A + 15 x^{2} A + 9 A + 5 x^{3} B + 3 x B + 5 C x^{3} + 3 C x^{2} + 5 D x^{2} + 3 D x$

Step 1.8.6

Move $9 A$ .

$1 = 25 x^{3} A + 15 x A + 15 x^{2} A + 5 x^{3} B + 3 x B + 5 C x^{3} + 3 C x^{2} + 5 D x^{2} + 3 D x + 9 A$

Step 1.8.7

Move $3 x B$ .

$1 = 25 x^{3} A + 15 x A + 15 x^{2} A + 5 x^{3} B + 5 C x^{3} + 3 C x^{2} + 5 D x^{2} + 3 x B + 3 D x + 9 A$

Step 1.8.8

Move $15 x A$ .

$1 = 25 x^{3} A + 15 x^{2} A + 5 x^{3} B + 5 C x^{3} + 3 C x^{2} + 5 D x^{2} + 15 x A + 3 x B + 3 D x + 9 A$

Step 1.8.9

Move $15 x^{2} A$ .

$1 = 25 x^{3} A + 5 x^{3} B + 5 C x^{3} + 15 x^{2} A + 3 C x^{2} + 5 D x^{2} + 15 x A + 3 x B + 3 D x + 9 A$

Step 2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 2.1

Create an equation for the partial fraction variables by equating the coefficients of $x^{3}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = 25 A + 5 B + 5 C$

Step 2.2

Create an equation for the partial fraction variables by equating the coefficients of $x^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = 15 A + 3 C + 5 D$

Step 2.3

Create an equation for the partial fraction variables by equating the coefficients of $x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = 15 A + 3 B + 3 D$

Step 2.4

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = 9 A$

Step 2.5

Set up the system of equations to find the coefficients of the partial fractions.

$0 = 25 A + 5 B + 5 C$

$0 = 15 A + 3 C + 5 D$

$0 = 15 A + 3 B + 3 D$

$1 = 9 A$

$0 = 25 A + 5 B + 5 C$

$0 = 15 A + 3 C + 5 D$

$0 = 15 A + 3 B + 3 D$

$1 = 9 A$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $1 = 9 A$ .

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Step 3.1.1

Rewrite the equation as $9 A = 1$ .

$9 A = 1$

$0 = 25 A + 5 B + 5 C$

$0 = 15 A + 3 C + 5 D$

$0 = 15 A + 3 B + 3 D$

Step 3.1.2

Divide each term in $9 A = 1$ by $9$ and simplify.

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Step 3.1.2.1

Divide each term in $9 A = 1$ by $9$ .

$\frac{9 A}{9} = \frac{1}{9}$

$0 = 25 A + 5 B + 5 C$

$0 = 15 A + 3 C + 5 D$

$0 = 15 A + 3 B + 3 D$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $9$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{9 A}{9} = \frac{1}{9}$

$0 = 25 A + 5 B + 5 C$

$0 = 15 A + 3 C + 5 D$

$0 = 15 A + 3 B + 3 D$

Step 3.1.2.2.1.2

Divide $A$ by $1$ .

$A = \frac{1}{9}$

$0 = 25 A + 5 B + 5 C$

$0 = 15 A + 3 C + 5 D$

$0 = 15 A + 3 B + 3 D$

$A = \frac{1}{9}$

$0 = 25 A + 5 B + 5 C$

$0 = 15 A + 3 C + 5 D$

$0 = 15 A + 3 B + 3 D$

$A = \frac{1}{9}$

$0 = 25 A + 5 B + 5 C$

$0 = 15 A + 3 C + 5 D$

$0 = 15 A + 3 B + 3 D$

$A = \frac{1}{9}$

$0 = 25 A + 5 B + 5 C$

$0 = 15 A + 3 C + 5 D$

$0 = 15 A + 3 B + 3 D$

$A = \frac{1}{9}$

$0 = 25 A + 5 B + 5 C$

$0 = 15 A + 3 C + 5 D$

$0 = 15 A + 3 B + 3 D$

Step 3.2

Replace all occurrences of $A$ with $\frac{1}{9}$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $0 = 25 A + 5 B + 5 C$ with $\frac{1}{9}$ .

$0 = 25 (\frac{1}{9}) + 5 B + 5 C$

$A = \frac{1}{9}$

$0 = 15 A + 3 C + 5 D$

$0 = 15 A + 3 B + 3 D$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Combine $25$ and $\frac{1}{9}$ .

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$0 = 15 A + 3 C + 5 D$

$0 = 15 A + 3 B + 3 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$0 = 15 A + 3 C + 5 D$

$0 = 15 A + 3 B + 3 D$

Step 3.2.3

Replace all occurrences of $A$ in $0 = 15 A + 3 C + 5 D$ with $\frac{1}{9}$ .

$0 = 15 (\frac{1}{9}) + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$0 = 15 A + 3 B + 3 D$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

Cancel the common factor of $3$ .

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Step 3.2.4.1.1.1

Factor $3$ out of $15$ .

$0 = 3 (5) (\frac{1}{9}) + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$0 = 15 A + 3 B + 3 D$

Step 3.2.4.1.1.2

Factor $3$ out of $9$ .

$0 = 3 \cdot (5 (\frac{1}{3 \cdot 3})) + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$0 = 15 A + 3 B + 3 D$

Step 3.2.4.1.1.3

Cancel the common factor.

$0 = 3 \cdot (5 (\frac{1}{3 \cdot 3})) + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$0 = 15 A + 3 B + 3 D$

Step 3.2.4.1.1.4

Rewrite the expression.

$0 = 5 (\frac{1}{3}) + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$0 = 15 A + 3 B + 3 D$

$0 = 5 (\frac{1}{3}) + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$0 = 15 A + 3 B + 3 D$

Step 3.2.4.1.2

Combine $5$ and $\frac{1}{3}$ .

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$0 = 15 A + 3 B + 3 D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$0 = 15 A + 3 B + 3 D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$0 = 15 A + 3 B + 3 D$

Step 3.2.5

Replace all occurrences of $A$ in $0 = 15 A + 3 B + 3 D$ with $\frac{1}{9}$ .

$0 = 15 (\frac{1}{9}) + 3 B + 3 D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.2.6

Simplify the right side.

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Step 3.2.6.1

Simplify each term.

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Step 3.2.6.1.1

Cancel the common factor of $3$ .

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Step 3.2.6.1.1.1

Factor $3$ out of $15$ .

$0 = 3 (5) (\frac{1}{9}) + 3 B + 3 D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.2.6.1.1.2

Factor $3$ out of $9$ .

$0 = 3 \cdot (5 (\frac{1}{3 \cdot 3})) + 3 B + 3 D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.2.6.1.1.3

Cancel the common factor.

$0 = 3 \cdot (5 (\frac{1}{3 \cdot 3})) + 3 B + 3 D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.2.6.1.1.4

Rewrite the expression.

$0 = 5 (\frac{1}{3}) + 3 B + 3 D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$0 = 5 (\frac{1}{3}) + 3 B + 3 D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.2.6.1.2

Combine $5$ and $\frac{1}{3}$ .

$0 = \frac{5}{3} + 3 B + 3 D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$0 = \frac{5}{3} + 3 B + 3 D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$0 = \frac{5}{3} + 3 B + 3 D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$0 = \frac{5}{3} + 3 B + 3 D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.3

Solve for $B$ in $0 = \frac{5}{3} + 3 B + 3 D$ .

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Step 3.3.1

Rewrite the equation as $\frac{5}{3} + 3 B + 3 D = 0$ .

$\frac{5}{3} + 3 B + 3 D = 0$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.3.2

Move all terms not containing $B$ to the right side of the equation.

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Step 3.3.2.1

Subtract $\frac{5}{3}$ from both sides of the equation.

$3 B + 3 D = - \frac{5}{3}$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.3.2.2

Subtract $3 D$ from both sides of the equation.

$3 B = - \frac{5}{3} - 3 D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$3 B = - \frac{5}{3} - 3 D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.3.3

Divide each term in $3 B = - \frac{5}{3} - 3 D$ by $3$ and simplify.

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Step 3.3.3.1

Divide each term in $3 B = - \frac{5}{3} - 3 D$ by $3$ .

$\frac{3 B}{3} = \frac{- \frac{5}{3}}{3} + \frac{- 3 D}{3}$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{3 B}{3} = \frac{- \frac{5}{3}}{3} + \frac{- 3 D}{3}$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.3.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{- \frac{5}{3}}{3} + \frac{- 3 D}{3}$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$B = \frac{- \frac{5}{3}}{3} + \frac{- 3 D}{3}$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$B = \frac{- \frac{5}{3}}{3} + \frac{- 3 D}{3}$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Simplify each term.

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Step 3.3.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$B = - \frac{5}{3} \cdot \frac{1}{3} + \frac{- 3 D}{3}$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.3.3.3.1.2

Multiply $- \frac{5}{3} \cdot \frac{1}{3}$ .

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Step 3.3.3.3.1.2.1

Multiply $\frac{1}{3}$ by $\frac{5}{3}$ .

$B = - \frac{5}{3 \cdot 3} + \frac{- 3 D}{3}$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.3.3.3.1.2.2

Multiply $3$ by $3$ .

$B = - \frac{5}{9} + \frac{- 3 D}{3}$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$B = - \frac{5}{9} + \frac{- 3 D}{3}$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.3.3.3.1.3

Cancel the common factor of $- 3$ and $3$ .

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Step 3.3.3.3.1.3.1

Factor $3$ out of $- 3 D$ .

$B = - \frac{5}{9} + \frac{3 (- D)}{3}$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.3.3.3.1.3.2

Cancel the common factors.

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Step 3.3.3.3.1.3.2.1

Factor $3$ out of $3$ .

$B = - \frac{5}{9} + \frac{3 (- D)}{3 (1)}$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.3.3.3.1.3.2.2

Cancel the common factor.

$B = - \frac{5}{9} + \frac{3 (- D)}{3 \cdot 1}$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.3.3.3.1.3.2.3

Rewrite the expression.

$B = - \frac{5}{9} + \frac{- D}{1}$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.3.3.3.1.3.2.4

Divide $- D$ by $1$ .

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$0 = \frac{25}{9} + 5 B + 5 C$

$A = \frac{1}{9}$

Step 3.4

Replace all occurrences of $B$ with $- \frac{5}{9} - D$ in each equation.

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Step 3.4.1

Replace all occurrences of $B$ in $0 = \frac{25}{9} + 5 B + 5 C$ with $- \frac{5}{9} - D$ .

$0 = \frac{25}{9} + 5 (- \frac{5}{9} - D) + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $\frac{25}{9} + 5 (- \frac{5}{9} - D) + 5 C$ .

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Step 3.4.2.1.1

Simplify each term.

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Step 3.4.2.1.1.1

Apply the distributive property.

$0 = \frac{25}{9} + 5 (- \frac{5}{9}) + 5 (- D) + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.4.2.1.1.2

Multiply $5 (- \frac{5}{9})$ .

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Step 3.4.2.1.1.2.1

Multiply $- 1$ by $5$ .

$0 = \frac{25}{9} - 5 (\frac{5}{9}) + 5 (- D) + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.4.2.1.1.2.2

Combine $- 5$ and $\frac{5}{9}$ .

$0 = \frac{25}{9} + \frac{- 5 \cdot 5}{9} + 5 (- D) + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.4.2.1.1.2.3

Multiply $- 5$ by $5$ .

$0 = \frac{25}{9} + \frac{- 25}{9} + 5 (- D) + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

$0 = \frac{25}{9} + \frac{- 25}{9} + 5 (- D) + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.4.2.1.1.3

Multiply $- 1$ by $5$ .

$0 = \frac{25}{9} + \frac{- 25}{9} - 5 D + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.4.2.1.1.4

Move the negative in front of the fraction.

$0 = \frac{25}{9} - \frac{25}{9} - 5 D + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

$0 = \frac{25}{9} - \frac{25}{9} - 5 D + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.4.2.1.2

Simplify by adding terms.

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Step 3.4.2.1.2.1

Combine the opposite terms in $\frac{25}{9} - \frac{25}{9} - 5 D + 5 C$ .

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Step 3.4.2.1.2.1.1

Combine the numerators over the common denominator.

$0 = \frac{25 - 25}{9} - 5 D + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.4.2.1.2.1.2

Subtract $25$ from $25$ .

$0 = \frac{0}{9} - 5 D + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.4.2.1.2.1.3

Divide $0$ by $9$ .

$0 = 0 - 5 D + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

$0 = 0 - 5 D + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.4.2.1.2.2

Subtract $5 D$ from $0$ .

$0 = - 5 D + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

$0 = - 5 D + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

$0 = - 5 D + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

$0 = - 5 D + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

$0 = - 5 D + 5 C$

$B = - \frac{5}{9} - D$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.5

Reorder $- \frac{5}{9}$ and $- D$ .

$B = - D - \frac{5}{9}$

$0 = - 5 D + 5 C$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.6

Solve for $D$ in $B = - D - \frac{5}{9}$ .

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Step 3.6.1

Rewrite the equation as $- D - \frac{5}{9} = B$ .

$- D - \frac{5}{9} = B$

$0 = - 5 D + 5 C$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.6.2

Add $\frac{5}{9}$ to both sides of the equation.

$- D = B + \frac{5}{9}$

$0 = - 5 D + 5 C$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.6.3

Divide each term in $- D = B + \frac{5}{9}$ by $- 1$ and simplify.

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Step 3.6.3.1

Divide each term in $- D = B + \frac{5}{9}$ by $- 1$ .

$\frac{- D}{- 1} = \frac{B}{- 1} + \frac{\frac{5}{9}}{- 1}$

$0 = - 5 D + 5 C$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.6.3.2

Simplify the left side.

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Step 3.6.3.2.1

Dividing two negative values results in a positive value.

$\frac{D}{1} = \frac{B}{- 1} + \frac{\frac{5}{9}}{- 1}$

$0 = - 5 D + 5 C$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.6.3.2.2

Divide $D$ by $1$ .

$D = \frac{B}{- 1} + \frac{\frac{5}{9}}{- 1}$

$0 = - 5 D + 5 C$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

$D = \frac{B}{- 1} + \frac{\frac{5}{9}}{- 1}$

$0 = - 5 D + 5 C$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.6.3.3

Simplify the right side.

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Step 3.6.3.3.1

Simplify each term.

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Step 3.6.3.3.1.1

Move the negative one from the denominator of $\frac{B}{- 1}$ .

$D = - 1 \cdot B + \frac{\frac{5}{9}}{- 1}$

$0 = - 5 D + 5 C$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.6.3.3.1.2

Rewrite $- 1 \cdot B$ as $- B$ .

$D = - B + \frac{\frac{5}{9}}{- 1}$

$0 = - 5 D + 5 C$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.6.3.3.1.3

Move the negative one from the denominator of $\frac{\frac{5}{9}}{- 1}$ .

$D = - B - 1 \cdot \frac{5}{9}$

$0 = - 5 D + 5 C$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.6.3.3.1.4

Rewrite $- 1 \cdot \frac{5}{9}$ as $- \frac{5}{9}$ .

$D = - B - \frac{5}{9}$

$0 = - 5 D + 5 C$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

$D = - B - \frac{5}{9}$

$0 = - 5 D + 5 C$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

$D = - B - \frac{5}{9}$

$0 = - 5 D + 5 C$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

$D = - B - \frac{5}{9}$

$0 = - 5 D + 5 C$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

$D = - B - \frac{5}{9}$

$0 = - 5 D + 5 C$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.7

Replace all occurrences of $D$ with $- B - \frac{5}{9}$ in each equation.

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Step 3.7.1

Replace all occurrences of $D$ in $0 = - 5 D + 5 C$ with $- B - \frac{5}{9}$ .

$0 = - 5 (- B - \frac{5}{9}) + 5 C$

$D = - B - \frac{5}{9}$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.7.2

Simplify the right side.

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Step 3.7.2.1

Simplify each term.

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Step 3.7.2.1.1

Apply the distributive property.

$0 = - 5 (- B) - 5 (- \frac{5}{9}) + 5 C$

$D = - B - \frac{5}{9}$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.7.2.1.2

Multiply $- 1$ by $- 5$ .

$0 = 5 B - 5 (- \frac{5}{9}) + 5 C$

$D = - B - \frac{5}{9}$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.7.2.1.3

Multiply $- 5 (- \frac{5}{9})$ .

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Step 3.7.2.1.3.1

Multiply $- 1$ by $- 5$ .

$0 = 5 B + 5 (\frac{5}{9}) + 5 C$

$D = - B - \frac{5}{9}$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.7.2.1.3.2

Combine $5$ and $\frac{5}{9}$ .

$0 = 5 B + \frac{5 \cdot 5}{9} + 5 C$

$D = - B - \frac{5}{9}$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.7.2.1.3.3

Multiply $5$ by $5$ .

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$0 = \frac{5}{3} + 3 C + 5 D$

$A = \frac{1}{9}$

Step 3.7.3

Replace all occurrences of $D$ in $0 = \frac{5}{3} + 3 C + 5 D$ with $- B - \frac{5}{9}$ .

$0 = \frac{5}{3} + 3 C + 5 (- B - \frac{5}{9})$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.7.4

Simplify the right side.

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Step 3.7.4.1

Simplify $\frac{5}{3} + 3 C + 5 (- B - \frac{5}{9})$ .

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Step 3.7.4.1.1

Simplify each term.

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Step 3.7.4.1.1.1

Apply the distributive property.

$0 = \frac{5}{3} + 3 C + 5 (- B) + 5 (- \frac{5}{9})$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.7.4.1.1.2

Multiply $- 1$ by $5$ .

$0 = \frac{5}{3} + 3 C - 5 B + 5 (- \frac{5}{9})$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.7.4.1.1.3

Multiply $5 (- \frac{5}{9})$ .

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Step 3.7.4.1.1.3.1

Multiply $- 1$ by $5$ .

$0 = \frac{5}{3} + 3 C - 5 B - 5 (\frac{5}{9})$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.7.4.1.1.3.2

Combine $- 5$ and $\frac{5}{9}$ .

$0 = \frac{5}{3} + 3 C - 5 B + \frac{- 5 \cdot 5}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.7.4.1.1.3.3

Multiply $- 5$ by $5$ .

$0 = \frac{5}{3} + 3 C - 5 B + \frac{- 25}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = \frac{5}{3} + 3 C - 5 B + \frac{- 25}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.7.4.1.1.4

Move the negative in front of the fraction.

$0 = \frac{5}{3} + 3 C - 5 B - \frac{25}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = \frac{5}{3} + 3 C - 5 B - \frac{25}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.7.4.1.2

To write $\frac{5}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$0 = 3 C - 5 B + \frac{5}{3} \cdot \frac{3}{3} - \frac{25}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.7.4.1.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.7.4.1.3.1

Multiply $\frac{5}{3}$ by $\frac{3}{3}$ .

$0 = 3 C - 5 B + \frac{5 \cdot 3}{3 \cdot 3} - \frac{25}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.7.4.1.3.2

Multiply $3$ by $3$ .

$0 = 3 C - 5 B + \frac{5 \cdot 3}{9} - \frac{25}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = 3 C - 5 B + \frac{5 \cdot 3}{9} - \frac{25}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.7.4.1.4

Combine the numerators over the common denominator.

$0 = 3 C - 5 B + \frac{5 \cdot 3 - 25}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.7.4.1.5

Simplify the numerator.

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Step 3.7.4.1.5.1

Multiply $5$ by $3$ .

$0 = 3 C - 5 B + \frac{15 - 25}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.7.4.1.5.2

Subtract $25$ from $15$ .

$0 = 3 C - 5 B + \frac{- 10}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = 3 C - 5 B + \frac{- 10}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.7.4.1.6

Move the negative in front of the fraction.

$0 = 3 C - 5 B - \frac{10}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = 3 C - 5 B - \frac{10}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = 3 C - 5 B - \frac{10}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = 3 C - 5 B - \frac{10}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.8

Solve for $C$ in $0 = 3 C - 5 B - \frac{10}{9}$ .

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Step 3.8.1

Rewrite the equation as $3 C - 5 B - \frac{10}{9} = 0$ .

$3 C - 5 B - \frac{10}{9} = 0$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.8.2

Move all terms not containing $C$ to the right side of the equation.

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Step 3.8.2.1

Add $5 B$ to both sides of the equation.

$3 C - \frac{10}{9} = 5 B$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.8.2.2

Add $\frac{10}{9}$ to both sides of the equation.

$3 C = 5 B + \frac{10}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$3 C = 5 B + \frac{10}{9}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.8.3

Divide each term in $3 C = 5 B + \frac{10}{9}$ by $3$ and simplify.

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Step 3.8.3.1

Divide each term in $3 C = 5 B + \frac{10}{9}$ by $3$ .

$\frac{3 C}{3} = \frac{5 B}{3} + \frac{\frac{10}{9}}{3}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.8.3.2

Simplify the left side.

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Step 3.8.3.2.1

Cancel the common factor of $3$ .

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Step 3.8.3.2.1.1

Cancel the common factor.

$\frac{3 C}{3} = \frac{5 B}{3} + \frac{\frac{10}{9}}{3}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.8.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{5 B}{3} + \frac{\frac{10}{9}}{3}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$C = \frac{5 B}{3} + \frac{\frac{10}{9}}{3}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$C = \frac{5 B}{3} + \frac{\frac{10}{9}}{3}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.8.3.3

Simplify the right side.

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Step 3.8.3.3.1

Simplify each term.

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Step 3.8.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$C = \frac{5 B}{3} + \frac{10}{9} \cdot \frac{1}{3}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.8.3.3.1.2

Multiply $\frac{10}{9} \cdot \frac{1}{3}$ .

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Step 3.8.3.3.1.2.1

Multiply $\frac{10}{9}$ by $\frac{1}{3}$ .

$C = \frac{5 B}{3} + \frac{10}{9 \cdot 3}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.8.3.3.1.2.2

Multiply $9$ by $3$ .

$C = \frac{5 B}{3} + \frac{10}{27}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$0 = 5 B + \frac{25}{9} + 5 C$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9

Replace all occurrences of $C$ with $\frac{5 B}{3} + \frac{10}{27}$ in each equation.

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Step 3.9.1

Replace all occurrences of $C$ in $0 = 5 B + \frac{25}{9} + 5 C$ with $\frac{5 B}{3} + \frac{10}{27}$ .

$0 = 5 B + \frac{25}{9} + 5 (\frac{5 B}{3} + \frac{10}{27})$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2

Simplify the right side.

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Step 3.9.2.1

Simplify $5 B + \frac{25}{9} + 5 (\frac{5 B}{3} + \frac{10}{27})$ .

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Step 3.9.2.1.1

Simplify each term.

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Step 3.9.2.1.1.1

Apply the distributive property.

$0 = 5 B + \frac{25}{9} + 5 (\frac{5 B}{3}) + 5 (\frac{10}{27})$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.1.2

Multiply $5 \frac{5 B}{3}$ .

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Step 3.9.2.1.1.2.1

Combine $5$ and $\frac{5 B}{3}$ .

$0 = 5 B + \frac{25}{9} + \frac{5 (5 B)}{3} + 5 (\frac{10}{27})$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.1.2.2

Multiply $5$ by $5$ .

$0 = 5 B + \frac{25}{9} + \frac{25 B}{3} + 5 (\frac{10}{27})$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = 5 B + \frac{25}{9} + \frac{25 B}{3} + 5 (\frac{10}{27})$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.1.3

Multiply $5 (\frac{10}{27})$ .

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Step 3.9.2.1.1.3.1

Combine $5$ and $\frac{10}{27}$ .

$0 = 5 B + \frac{25}{9} + \frac{25 B}{3} + \frac{5 \cdot 10}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.1.3.2

Multiply $5$ by $10$ .

$0 = 5 B + \frac{25}{9} + \frac{25 B}{3} + \frac{50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = 5 B + \frac{25}{9} + \frac{25 B}{3} + \frac{50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = 5 B + \frac{25}{9} + \frac{25 B}{3} + \frac{50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.2

To write $5 B$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$0 = 5 B \cdot \frac{3}{3} + \frac{25 B}{3} + \frac{25}{9} + \frac{50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.3

Combine $5 B$ and $\frac{3}{3}$ .

$0 = \frac{5 B \cdot 3}{3} + \frac{25 B}{3} + \frac{25}{9} + \frac{50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.4

Combine the numerators over the common denominator.

$0 = \frac{5 B \cdot 3 + 25 B}{3} + \frac{25}{9} + \frac{50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.5

Find the common denominator.

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Step 3.9.2.1.5.1

Multiply $\frac{5 B \cdot 3 + 25 B}{3}$ by $\frac{9}{9}$ .

$0 = \frac{5 B \cdot 3 + 25 B}{3} \cdot \frac{9}{9} + \frac{25}{9} + \frac{50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.5.2

Multiply $\frac{5 B \cdot 3 + 25 B}{3}$ by $\frac{9}{9}$ .

$0 = \frac{(5 B \cdot 3 + 25 B) \cdot 9}{3 \cdot 9} + \frac{25}{9} + \frac{50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.5.3

Multiply $\frac{25}{9}$ by $\frac{3}{3}$ .

$0 = \frac{(5 B \cdot 3 + 25 B) \cdot 9}{3 \cdot 9} + \frac{25}{9} \cdot \frac{3}{3} + \frac{50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.5.4

Multiply $\frac{25}{9}$ by $\frac{3}{3}$ .

$0 = \frac{(5 B \cdot 3 + 25 B) \cdot 9}{3 \cdot 9} + \frac{25 \cdot 3}{9 \cdot 3} + \frac{50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.5.5

Multiply $3$ by $9$ .

$0 = \frac{(5 B \cdot 3 + 25 B) \cdot 9}{27} + \frac{25 \cdot 3}{9 \cdot 3} + \frac{50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.5.6

Reorder the factors of $9 \cdot 3$ .

$0 = \frac{(5 B \cdot 3 + 25 B) \cdot 9}{27} + \frac{25 \cdot 3}{3 \cdot 9} + \frac{50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.5.7

Multiply $3$ by $9$ .

$0 = \frac{(5 B \cdot 3 + 25 B) \cdot 9}{27} + \frac{25 \cdot 3}{27} + \frac{50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = \frac{(5 B \cdot 3 + 25 B) \cdot 9}{27} + \frac{25 \cdot 3}{27} + \frac{50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.6

Combine the numerators over the common denominator.

$0 = \frac{(5 B \cdot 3 + 25 B) \cdot 9 + 25 \cdot 3 + 50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.7

Simplify each term.

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Step 3.9.2.1.7.1

Multiply $3$ by $5$ .

$0 = \frac{(15 B + 25 B) \cdot 9 + 25 \cdot 3 + 50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.7.2

Add $15 B$ and $25 B$ .

$0 = \frac{40 B \cdot 9 + 25 \cdot 3 + 50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.7.3

Multiply $9$ by $40$ .

$0 = \frac{360 B + 25 \cdot 3 + 50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.7.4

Multiply $25$ by $3$ .

$0 = \frac{360 B + 75 + 50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = \frac{360 B + 75 + 50}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.8

Simplify with factoring out.

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Step 3.9.2.1.8.1

Add $75$ and $50$ .

$0 = \frac{360 B + 125}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.8.2

Factor $5$ out of $360 B + 125$ .

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Step 3.9.2.1.8.2.1

Factor $5$ out of $360 B$ .

$0 = \frac{5 (72 B) + 125}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.8.2.2

Factor $5$ out of $125$ .

$0 = \frac{5 (72 B) + 5 (25)}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.9.2.1.8.2.3

Factor $5$ out of $5 (72 B) + 5 (25)$ .

$0 = \frac{5 (72 B + 25)}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = \frac{5 (72 B + 25)}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = \frac{5 (72 B + 25)}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = \frac{5 (72 B + 25)}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = \frac{5 (72 B + 25)}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$0 = \frac{5 (72 B + 25)}{27}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.10

Solve for $B$ in $0 = \frac{5 (72 B + 25)}{27}$ .

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Step 3.10.1

Set the numerator equal to zero.

$5 (72 B + 25) = 0$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.10.2

Solve the equation for $B$ .

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Step 3.10.2.1

Divide each term in $5 (72 B + 25) = 0$ by $5$ and simplify.

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Step 3.10.2.1.1

Divide each term in $5 (72 B + 25) = 0$ by $5$ .

$\frac{5 (72 B + 25)}{5} = \frac{0}{5}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.10.2.1.2

Simplify the left side.

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Step 3.10.2.1.2.1

Cancel the common factor of $5$ .

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Step 3.10.2.1.2.1.1

Cancel the common factor.

$\frac{5 (72 B + 25)}{5} = \frac{0}{5}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.10.2.1.2.1.2

Divide $72 B + 25$ by $1$ .

$72 B + 25 = \frac{0}{5}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$72 B + 25 = \frac{0}{5}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$72 B + 25 = \frac{0}{5}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.10.2.1.3

Simplify the right side.

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Step 3.10.2.1.3.1

Divide $0$ by $5$ .

$72 B + 25 = 0$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$72 B + 25 = 0$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$72 B + 25 = 0$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.10.2.2

Subtract $25$ from both sides of the equation.

$72 B = - 25$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.10.2.3

Divide each term in $72 B = - 25$ by $72$ and simplify.

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Step 3.10.2.3.1

Divide each term in $72 B = - 25$ by $72$ .

$\frac{72 B}{72} = \frac{- 25}{72}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.10.2.3.2

Simplify the left side.

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Step 3.10.2.3.2.1

Cancel the common factor of $72$ .

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Step 3.10.2.3.2.1.1

Cancel the common factor.

$\frac{72 B}{72} = \frac{- 25}{72}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.10.2.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{- 25}{72}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$B = \frac{- 25}{72}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$B = \frac{- 25}{72}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.10.2.3.3

Simplify the right side.

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Step 3.10.2.3.3.1

Move the negative in front of the fraction.

$B = - \frac{25}{72}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$B = - \frac{25}{72}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$B = - \frac{25}{72}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$B = - \frac{25}{72}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$B = - \frac{25}{72}$

$C = \frac{5 B}{3} + \frac{10}{27}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11

Replace all occurrences of $B$ with $- \frac{25}{72}$ in each equation.

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Step 3.11.1

Replace all occurrences of $B$ in $C = \frac{5 B}{3} + \frac{10}{27}$ with $- \frac{25}{72}$ .

$C = \frac{5 (- \frac{25}{72})}{3} + \frac{10}{27}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2

Simplify the right side.

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Step 3.11.2.1

Simplify $\frac{5 (- \frac{25}{72})}{3} + \frac{10}{27}$ .

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Step 3.11.2.1.1

Simplify each term.

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Step 3.11.2.1.1.1

Simplify the numerator.

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Step 3.11.2.1.1.1.1

Multiply $- 1$ by $5$ .

$C = \frac{- 5 (\frac{25}{72})}{3} + \frac{10}{27}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.1.1.2

Combine $- 5$ and $\frac{25}{72}$ .

$C = \frac{\frac{- 5 \cdot 25}{72}}{3} + \frac{10}{27}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$C = \frac{\frac{- 5 \cdot 25}{72}}{3} + \frac{10}{27}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.1.2

Multiply $- 5$ by $25$ .

$C = \frac{\frac{- 125}{72}}{3} + \frac{10}{27}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.1.3

Move the negative in front of the fraction.

$C = \frac{- \frac{125}{72}}{3} + \frac{10}{27}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.1.4

Multiply the numerator by the reciprocal of the denominator.

$C = - \frac{125}{72} \cdot \frac{1}{3} + \frac{10}{27}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.1.5

Multiply $- \frac{125}{72} \cdot \frac{1}{3}$ .

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Step 3.11.2.1.1.5.1

Multiply $\frac{1}{3}$ by $\frac{125}{72}$ .

$C = - \frac{125}{3 \cdot 72} + \frac{10}{27}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.1.5.2

Multiply $3$ by $72$ .

$C = - \frac{125}{216} + \frac{10}{27}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$C = - \frac{125}{216} + \frac{10}{27}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$C = - \frac{125}{216} + \frac{10}{27}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.2

To write $\frac{10}{27}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$C = - \frac{125}{216} + \frac{10}{27} \cdot \frac{8}{8}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.3

Write each expression with a common denominator of $216$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.11.2.1.3.1

Multiply $\frac{10}{27}$ by $\frac{8}{8}$ .

$C = - \frac{125}{216} + \frac{10 \cdot 8}{27 \cdot 8}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.3.2

Multiply $27$ by $8$ .

$C = - \frac{125}{216} + \frac{10 \cdot 8}{216}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$C = - \frac{125}{216} + \frac{10 \cdot 8}{216}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.4

Combine the numerators over the common denominator.

$C = \frac{- 125 + 10 \cdot 8}{216}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.5

Simplify the numerator.

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Step 3.11.2.1.5.1

Multiply $10$ by $8$ .

$C = \frac{- 125 + 80}{216}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.5.2

Add $- 125$ and $80$ .

$C = \frac{- 45}{216}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$C = \frac{- 45}{216}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.6

Cancel the common factor of $- 45$ and $216$ .

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Step 3.11.2.1.6.1

Factor $9$ out of $- 45$ .

$C = \frac{9 (- 5)}{216}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.6.2

Cancel the common factors.

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Step 3.11.2.1.6.2.1

Factor $9$ out of $216$ .

$C = \frac{9 \cdot - 5}{9 \cdot 24}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.6.2.2

Cancel the common factor.

$C = \frac{9 \cdot - 5}{9 \cdot 24}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.6.2.3

Rewrite the expression.

$C = \frac{- 5}{24}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$C = \frac{- 5}{24}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$C = \frac{- 5}{24}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.2.1.7

Move the negative in front of the fraction.

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$D = - B - \frac{5}{9}$

$A = \frac{1}{9}$

Step 3.11.3

Replace all occurrences of $B$ in $D = - B - \frac{5}{9}$ with $- \frac{25}{72}$ .

$D = - (- \frac{25}{72}) - \frac{5}{9}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

Step 3.11.4

Simplify the right side.

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Step 3.11.4.1

Simplify $- (- \frac{25}{72}) - \frac{5}{9}$ .

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Step 3.11.4.1.1

Multiply $- (- \frac{25}{72})$ .

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Step 3.11.4.1.1.1

Multiply $- 1$ by $- 1$ .

$D = 1 (\frac{25}{72}) - \frac{5}{9}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

Step 3.11.4.1.1.2

Multiply $\frac{25}{72}$ by $1$ .

$D = \frac{25}{72} - \frac{5}{9}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

$D = \frac{25}{72} - \frac{5}{9}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

Step 3.11.4.1.2

To write $- \frac{5}{9}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$D = \frac{25}{72} - \frac{5}{9} \cdot \frac{8}{8}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

Step 3.11.4.1.3

Write each expression with a common denominator of $72$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.11.4.1.3.1

Multiply $\frac{5}{9}$ by $\frac{8}{8}$ .

$D = \frac{25}{72} - \frac{5 \cdot 8}{9 \cdot 8}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

Step 3.11.4.1.3.2

Multiply $9$ by $8$ .

$D = \frac{25}{72} - \frac{5 \cdot 8}{72}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

$D = \frac{25}{72} - \frac{5 \cdot 8}{72}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

Step 3.11.4.1.4

Combine the numerators over the common denominator.

$D = \frac{25 - 5 \cdot 8}{72}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

Step 3.11.4.1.5

Simplify the numerator.

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Step 3.11.4.1.5.1

Multiply $- 5$ by $8$ .

$D = \frac{25 - 40}{72}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

Step 3.11.4.1.5.2

Subtract $40$ from $25$ .

$D = \frac{- 15}{72}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

$D = \frac{- 15}{72}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

Step 3.11.4.1.6

Cancel the common factor of $- 15$ and $72$ .

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Step 3.11.4.1.6.1

Factor $3$ out of $- 15$ .

$D = \frac{3 (- 5)}{72}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

Step 3.11.4.1.6.2

Cancel the common factors.

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Step 3.11.4.1.6.2.1

Factor $3$ out of $72$ .

$D = \frac{3 \cdot - 5}{3 \cdot 24}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

Step 3.11.4.1.6.2.2

Cancel the common factor.

$D = \frac{3 \cdot - 5}{3 \cdot 24}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

Step 3.11.4.1.6.2.3

Rewrite the expression.

$D = \frac{- 5}{24}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

$D = \frac{- 5}{24}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

$D = \frac{- 5}{24}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

Step 3.11.4.1.7

Move the negative in front of the fraction.

$D = - \frac{5}{24}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

$D = - \frac{5}{24}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

$D = - \frac{5}{24}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

$D = - \frac{5}{24}$

$C = - \frac{5}{24}$

$B = - \frac{25}{72}$

$A = \frac{1}{9}$

Step 3.12

List all of the solutions.

$D = - \frac{5}{24}, C = - \frac{5}{24}, B = - \frac{25}{72}, A = \frac{1}{9}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x} + \frac{B}{5 x + 3} + \frac{C x + D}{5 x^{2} + 3}$ with the values found for $A$ , $B$ , $C$ , and $D$ .

$\frac{\frac{1}{9}}{x} + \frac{- \frac{25}{72}}{5 x + 3} + \frac{- \frac{5}{24 x} - \frac{5}{24}}{5 x^{2} + 3}$

Step 5

Simplify each term.

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Step 5.1

Combine $x$ and $\frac{5}{24}$ .

$\frac{\frac{1}{9}}{x} + \frac{- \frac{25}{72}}{5 x + 3} + \frac{- \frac{x \cdot 5}{24} - \frac{5}{24}}{5 x^{2} + 3}$

Step 5.2

Move $5$ to the left of $x$ .

$\frac{\frac{1}{9}}{x} + \frac{- \frac{25}{72}}{5 x + 3} + \frac{- \frac{5 x}{24} - \frac{5}{24}}{5 x^{2} + 3}$