Algebra Examples

$\frac{2 x + 7}{x - 7} + \frac{x + 14}{x - 12} = \frac{x + 14}{x - 7}$

Step 1

Subtract $\frac{x + 14}{x - 7}$ from both sides of the equation.

$\frac{2 x + 7}{x - 7} + \frac{x + 14}{x - 12} - \frac{x + 14}{x - 7} = 0$

Step 2

Simplify $\frac{2 x + 7}{x - 7} + \frac{x + 14}{x - 12} - \frac{x + 14}{x - 7}$ .

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Step 2.1

Combine the numerators over the common denominator.

$\frac{2 x + 7 - (x + 14)}{x - 7} + \frac{x + 14}{x - 12} = 0$

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the distributive property.

$\frac{2 x + 7 - x - 1 \cdot 14}{x - 7} + \frac{x + 14}{x - 12} = 0$

Step 2.2.2

Multiply $- 1$ by $14$ .

$\frac{2 x + 7 - x - 14}{x - 7} + \frac{x + 14}{x - 12} = 0$

Step 2.3

Subtract $x$ from $2 x$ .

$\frac{x + 7 - 14}{x - 7} + \frac{x + 14}{x - 12} = 0$

Step 2.4

Subtract $14$ from $7$ .

$\frac{x - 7}{x - 7} + \frac{x + 14}{x - 12} = 0$

Step 2.5

Cancel the common factor of $x - 7$ .

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Step 2.5.1

Cancel the common factor.

$\frac{x - 7}{x - 7} + \frac{x + 14}{x - 12} = 0$

Step 2.5.2

Rewrite the expression.

$1 + \frac{x + 14}{x - 12} = 0$

Step 2.6

Write $1$ as a fraction with a common denominator.

$\frac{x - 12}{x - 12} + \frac{x + 14}{x - 12} = 0$

Step 2.7

Combine the numerators over the common denominator.

$\frac{x - 12 + x + 14}{x - 12} = 0$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Add $x$ and $x$ .

$\frac{2 x - 12 + 14}{x - 12} = 0$

Step 2.8.2

Add $- 12$ and $14$ .

$\frac{2 x + 2}{x - 12} = 0$

Step 2.8.3

Factor $2$ out of $2 x + 2$ .

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Step 2.8.3.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 2}{x - 12} = 0$

Step 2.8.3.2

Factor $2$ out of $2$ .

$\frac{2 (x) + 2 (1)}{x - 12} = 0$

Step 2.8.3.3

Factor $2$ out of $2 (x) + 2 (1)$ .

$\frac{2 (x + 1)}{x - 12} = 0$

Step 3

Set the numerator equal to zero.

$2 (x + 1) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Divide each term in $2 (x + 1) = 0$ by $2$ and simplify.

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Step 4.1.1

Divide each term in $2 (x + 1) = 0$ by $2$ .

$\frac{2 (x + 1)}{2} = \frac{0}{2}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{2 (x + 1)}{2} = \frac{0}{2}$

Step 4.1.2.1.2

Divide $x + 1$ by $1$ .

$x + 1 = \frac{0}{2}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $2$ .

$x + 1 = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$x = - 1$