Algebra Examples

$\frac{7}{3} - \frac{3}{x - 5} = \frac{19}{12}$

Step 1

Subtract $\frac{19}{12}$ from both sides of the equation.

$\frac{7}{3} - \frac{3}{x - 5} - \frac{19}{12} = 0$

Step 2

Simplify $\frac{7}{3} - \frac{3}{x - 5} - \frac{19}{12}$ .

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Step 2.1

Find the common denominator.

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Step 2.1.1

Multiply $\frac{7}{3}$ by $\frac{4 (x - 5)}{4 (x - 5)}$ .

$\frac{7}{3} \cdot \frac{4 (x - 5)}{4 (x - 5)} - \frac{3}{x - 5} - \frac{19}{12} = 0$

Step 2.1.2

Multiply $\frac{7}{3}$ by $\frac{4 (x - 5)}{4 (x - 5)}$ .

$\frac{7 (4 (x - 5))}{3 (4 (x - 5))} - \frac{3}{x - 5} - \frac{19}{12} = 0$

Step 2.1.3

Multiply $\frac{3}{x - 5}$ by $\frac{12}{12}$ .

$\frac{7 (4 (x - 5))}{3 (4 (x - 5))} - (\frac{3}{x - 5} \cdot \frac{12}{12}) - \frac{19}{12} = 0$

Step 2.1.4

Multiply $\frac{3}{x - 5}$ by $\frac{12}{12}$ .

$\frac{7 (4 (x - 5))}{3 (4 (x - 5))} - \frac{3 \cdot 12}{(x - 5) \cdot 12} - \frac{19}{12} = 0$

Step 2.1.5

Multiply $\frac{19}{12}$ by $\frac{x - 5}{x - 5}$ .

$\frac{7 (4 (x - 5))}{3 (4 (x - 5))} - \frac{3 \cdot 12}{(x - 5) \cdot 12} - (\frac{19}{12} \cdot \frac{x - 5}{x - 5}) = 0$

Step 2.1.6

Multiply $\frac{19}{12}$ by $\frac{x - 5}{x - 5}$ .

$\frac{7 (4 (x - 5))}{3 (4 (x - 5))} - \frac{3 \cdot 12}{(x - 5) \cdot 12} - \frac{19 (x - 5)}{12 (x - 5)} = 0$

Step 2.1.7

Reorder the factors of $3 (4 (x - 5))$ .

$\frac{7 (4 (x - 5))}{3 \cdot 4 (x - 5)} - \frac{3 \cdot 12}{(x - 5) \cdot 12} - \frac{19 (x - 5)}{12 (x - 5)} = 0$

Step 2.1.8

Multiply $3$ by $4$ .

$\frac{7 (4 (x - 5))}{12 (x - 5)} - \frac{3 \cdot 12}{(x - 5) \cdot 12} - \frac{19 (x - 5)}{12 (x - 5)} = 0$

Step 2.1.9

Reorder the factors of $(x - 5) \cdot 12$ .

$\frac{7 (4 (x - 5))}{12 (x - 5)} - \frac{3 \cdot 12}{12 (x - 5)} - \frac{19 (x - 5)}{12 (x - 5)} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{7 (4 (x - 5)) - 3 \cdot 12 - 19 (x - 5)}{12 (x - 5)} = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

$\frac{7 (4 x + 4 \cdot - 5) - 3 \cdot 12 - 19 (x - 5)}{12 (x - 5)} = 0$

Step 2.3.2

Multiply $4$ by $- 5$ .

$\frac{7 (4 x - 20) - 3 \cdot 12 - 19 (x - 5)}{12 (x - 5)} = 0$

Step 2.3.3

Apply the distributive property.

$\frac{7 (4 x) + 7 \cdot - 20 - 3 \cdot 12 - 19 (x - 5)}{12 (x - 5)} = 0$

Step 2.3.4

Multiply $4$ by $7$ .

$\frac{28 x + 7 \cdot - 20 - 3 \cdot 12 - 19 (x - 5)}{12 (x - 5)} = 0$

Step 2.3.5

Multiply $7$ by $- 20$ .

$\frac{28 x - 140 - 3 \cdot 12 - 19 (x - 5)}{12 (x - 5)} = 0$

Step 2.3.6

Multiply $- 3$ by $12$ .

$\frac{28 x - 140 - 36 - 19 (x - 5)}{12 (x - 5)} = 0$

Step 2.3.7

Apply the distributive property.

$\frac{28 x - 140 - 36 - 19 x - 19 \cdot - 5}{12 (x - 5)} = 0$

Step 2.3.8

Multiply $- 19$ by $- 5$ .

$\frac{28 x - 140 - 36 - 19 x + 95}{12 (x - 5)} = 0$

Step 2.4

Subtract $19 x$ from $28 x$ .

$\frac{9 x - 140 - 36 + 95}{12 (x - 5)} = 0$

Step 2.5

Subtract $36$ from $- 140$ .

$\frac{9 x - 176 + 95}{12 (x - 5)} = 0$

Step 2.6

Add $- 176$ and $95$ .

$\frac{9 x - 81}{12 (x - 5)} = 0$

Step 2.7

Cancel the common factor of $9 x - 81$ and $12$ .

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Step 2.7.1

Factor $3$ out of $9 x$ .

$\frac{3 (3 x) - 81}{12 (x - 5)} = 0$

Step 2.7.2

Factor $3$ out of $- 81$ .

$\frac{3 (3 x) + 3 (- 27)}{12 (x - 5)} = 0$

Step 2.7.3

Factor $3$ out of $3 (3 x) + 3 (- 27)$ .

$\frac{3 (3 x - 27)}{12 (x - 5)} = 0$

Step 2.7.4

Cancel the common factors.

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Step 2.7.4.1

Factor $3$ out of $12 (x - 5)$ .

$\frac{3 (3 x - 27)}{3 (4 (x - 5))} = 0$

Step 2.7.4.2

Cancel the common factor.

$\frac{3 (3 x - 27)}{3 (4 (x - 5))} = 0$

Step 2.7.4.3

Rewrite the expression.

$\frac{3 x - 27}{4 (x - 5)} = 0$

Step 2.8

Factor $3$ out of $3 x - 27$ .

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Step 2.8.1

Factor $3$ out of $3 x$ .

$\frac{3 (x) - 27}{4 (x - 5)} = 0$

Step 2.8.2

Factor $3$ out of $- 27$ .

$\frac{3 x + 3 \cdot - 9}{4 (x - 5)} = 0$

Step 2.8.3

Factor $3$ out of $3 x + 3 \cdot - 9$ .

$\frac{3 (x - 9)}{4 (x - 5)} = 0$

Step 3

Set the numerator equal to zero.

$3 (x - 9) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Divide each term in $3 (x - 9) = 0$ by $3$ and simplify.

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Step 4.1.1

Divide each term in $3 (x - 9) = 0$ by $3$ .

$\frac{3 (x - 9)}{3} = \frac{0}{3}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $3$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{3 (x - 9)}{3} = \frac{0}{3}$

Step 4.1.2.1.2

Divide $x - 9$ by $1$ .

$x - 9 = \frac{0}{3}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $3$ .

$x - 9 = 0$

Step 4.2

Add $9$ to both sides of the equation.

$x = 9$