Algebra Examples

$\frac{18}{3 x - 15} + \frac{1}{x - 5} = \frac{7}{3}$

Step 1

Subtract $\frac{7}{3}$ from both sides of the equation.

$\frac{18}{3 x - 15} + \frac{1}{x - 5} - \frac{7}{3} = 0$

Step 2

Simplify $\frac{18}{3 x - 15} + \frac{1}{x - 5} - \frac{7}{3}$ .

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Step 2.1

Cancel the common factor of $18$ and $3 x - 15$ .

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Step 2.1.1

Factor $3$ out of $18$ .

$\frac{3 \cdot 6}{3 x - 15} + \frac{1}{x - 5} - \frac{7}{3} = 0$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $3$ out of $3 x$ .

$\frac{3 \cdot 6}{3 (x) - 15} + \frac{1}{x - 5} - \frac{7}{3} = 0$

Step 2.1.2.2

Factor $3$ out of $- 15$ .

$\frac{3 \cdot 6}{3 (x) + 3 (- 5)} + \frac{1}{x - 5} - \frac{7}{3} = 0$

Step 2.1.2.3

Factor $3$ out of $3 (x) + 3 (- 5)$ .

$\frac{3 \cdot 6}{3 (x - 5)} + \frac{1}{x - 5} - \frac{7}{3} = 0$

Step 2.1.2.4

Cancel the common factor.

$\frac{3 \cdot 6}{3 (x - 5)} + \frac{1}{x - 5} - \frac{7}{3} = 0$

Step 2.1.2.5

Rewrite the expression.

$\frac{6}{x - 5} + \frac{1}{x - 5} - \frac{7}{3} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{6 + 1}{x - 5} + \frac{- 7}{3} = 0$

Step 2.3

Add $6$ and $1$ .

$\frac{7}{x - 5} + \frac{- 7}{3} = 0$

Step 2.4

Move the negative in front of the fraction.

$\frac{7}{x - 5} - \frac{7}{3} = 0$

Step 2.5

To write $\frac{7}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{7}{x - 5} \cdot \frac{3}{3} - \frac{7}{3} = 0$

Step 2.6

To write $- \frac{7}{3}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{7}{x - 5} \cdot \frac{3}{3} - \frac{7}{3} \cdot \frac{x - 5}{x - 5} = 0$

Step 2.7

Write each expression with a common denominator of $(x - 5) \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.7.1

Multiply $\frac{7}{x - 5}$ by $\frac{3}{3}$ .

$\frac{7 \cdot 3}{(x - 5) \cdot 3} - \frac{7}{3} \cdot \frac{x - 5}{x - 5} = 0$

Step 2.7.2

Multiply $\frac{7}{3}$ by $\frac{x - 5}{x - 5}$ .

$\frac{7 \cdot 3}{(x - 5) \cdot 3} - \frac{7 (x - 5)}{3 (x - 5)} = 0$

Step 2.7.3

Reorder the factors of $(x - 5) \cdot 3$ .

$\frac{7 \cdot 3}{3 (x - 5)} - \frac{7 (x - 5)}{3 (x - 5)} = 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{7 \cdot 3 - 7 (x - 5)}{3 (x - 5)} = 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Factor $7$ out of $7 \cdot 3 - 7 (x - 5)$ .

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Step 2.9.1.1

Factor $7$ out of $7 \cdot 3$ .

$\frac{7 (3) - 7 (x - 5)}{3 (x - 5)} = 0$

Step 2.9.1.2

Factor $7$ out of $- 7 (x - 5)$ .

$\frac{7 (3) + 7 (- (x - 5))}{3 (x - 5)} = 0$

Step 2.9.1.3

Factor $7$ out of $7 (3) + 7 (- (x - 5))$ .

$\frac{7 (3 - (x - 5))}{3 (x - 5)} = 0$

Step 2.9.2

Apply the distributive property.

$\frac{7 (3 - x - - 5)}{3 (x - 5)} = 0$

Step 2.9.3

Multiply $- 1$ by $- 5$ .

$\frac{7 (3 - x + 5)}{3 (x - 5)} = 0$

Step 2.9.4

Add $3$ and $5$ .

$\frac{7 (- x + 8)}{3 (x - 5)} = 0$

Step 2.10

Factor $- 1$ out of $- x$ .

$\frac{7 (- (x) + 8)}{3 (x - 5)} = 0$

Step 2.11

Rewrite $8$ as $- 1 (- 8)$ .

$\frac{7 (- (x) - 1 (- 8))}{3 (x - 5)} = 0$

Step 2.12

Factor $- 1$ out of $- (x) - 1 (- 8)$ .

$\frac{7 (- (x - 8))}{3 (x - 5)} = 0$

Step 2.13

Rewrite $- (x - 8)$ as $- 1 (x - 8)$ .

$\frac{7 (- 1 (x - 8))}{3 (x - 5)} = 0$

Step 2.14

Move the negative in front of the fraction.

$- \frac{7 (x - 8)}{3 (x - 5)} = 0$

Step 3

Set the numerator equal to zero.

$7 (x - 8) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Divide each term in $7 (x - 8) = 0$ by $7$ and simplify.

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Step 4.1.1

Divide each term in $7 (x - 8) = 0$ by $7$ .

$\frac{7 (x - 8)}{7} = \frac{0}{7}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $7$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{7 (x - 8)}{7} = \frac{0}{7}$

Step 4.1.2.1.2

Divide $x - 8$ by $1$ .

$x - 8 = \frac{0}{7}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $7$ .

$x - 8 = 0$

Step 4.2

Add $8$ to both sides of the equation.

$x = 8$