Algebra Examples

$\frac{2}{x - 2} + \frac{3}{x + 5} = \frac{r x + t}{(x - 2) (x + 5)}$

Step 1

Rewrite the equation as $\frac{r x + t}{(x - 2) (x + 5)} = \frac{2}{x - 2} + \frac{3}{x + 5}$ .

$\frac{r x + t}{(x - 2) (x + 5)} = \frac{2}{x - 2} + \frac{3}{x + 5}$

Step 2

Multiply both sides by $(x - 2) (x + 5)$ .

$\frac{r x + t}{(x - 2) (x + 5)} ((x - 2) (x + 5)) = (\frac{2}{x - 2} + \frac{3}{x + 5}) ((x - 2) (x + 5))$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $(x - 2) (x + 5)$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{r x + t}{(x - 2) (x + 5)} ((x - 2) (x + 5)) = (\frac{2}{x - 2} + \frac{3}{x + 5}) ((x - 2) (x + 5))$

Step 3.1.1.2

Rewrite the expression.

$r x + t = (\frac{2}{x - 2} + \frac{3}{x + 5}) ((x - 2) (x + 5))$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $(\frac{2}{x - 2} + \frac{3}{x + 5}) ((x - 2) (x + 5))$ .

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Step 3.2.1.1

To write $\frac{2}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$r x + t = (\frac{2}{x - 2} \cdot \frac{x + 5}{x + 5} + \frac{3}{x + 5}) ((x - 2) (x + 5))$

Step 3.2.1.2

To write $\frac{3}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$r x + t = (\frac{2}{x - 2} \cdot \frac{x + 5}{x + 5} + \frac{3}{x + 5} \cdot \frac{x - 2}{x - 2}) ((x - 2) (x + 5))$

Step 3.2.1.3

Write each expression with a common denominator of $(x - 2) (x + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.1.3.1

Multiply $\frac{2}{x - 2}$ by $\frac{x + 5}{x + 5}$ .

$r x + t = (\frac{2 (x + 5)}{(x - 2) (x + 5)} + \frac{3}{x + 5} \cdot \frac{x - 2}{x - 2}) ((x - 2) (x + 5))$

Step 3.2.1.3.2

Multiply $\frac{3}{x + 5}$ by $\frac{x - 2}{x - 2}$ .

$r x + t = (\frac{2 (x + 5)}{(x - 2) (x + 5)} + \frac{3 (x - 2)}{(x + 5) (x - 2)}) ((x - 2) (x + 5))$

Step 3.2.1.3.3

Reorder the factors of $(x - 2) (x + 5)$ .

$r x + t = (\frac{2 (x + 5)}{(x + 5) (x - 2)} + \frac{3 (x - 2)}{(x + 5) (x - 2)}) ((x - 2) (x + 5))$

Step 3.2.1.4

Combine the numerators over the common denominator.

$r x + t = \frac{2 (x + 5) + 3 (x - 2)}{(x + 5) (x - 2)} ((x - 2) (x + 5))$

Step 3.2.1.5

Simplify the numerator.

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Step 3.2.1.5.1

Apply the distributive property.

$r x + t = \frac{2 x + 2 \cdot 5 + 3 (x - 2)}{(x + 5) (x - 2)} ((x - 2) (x + 5))$

Step 3.2.1.5.2

Multiply $2$ by $5$ .

$r x + t = \frac{2 x + 10 + 3 (x - 2)}{(x + 5) (x - 2)} ((x - 2) (x + 5))$

Step 3.2.1.5.3

Apply the distributive property.

$r x + t = \frac{2 x + 10 + 3 x + 3 \cdot - 2}{(x + 5) (x - 2)} ((x - 2) (x + 5))$

Step 3.2.1.5.4

Multiply $3$ by $- 2$ .

$r x + t = \frac{2 x + 10 + 3 x - 6}{(x + 5) (x - 2)} ((x - 2) (x + 5))$

Step 3.2.1.5.5

Add $2 x$ and $3 x$ .

$r x + t = \frac{5 x + 10 - 6}{(x + 5) (x - 2)} ((x - 2) (x + 5))$

Step 3.2.1.5.6

Subtract $6$ from $10$ .

$r x + t = \frac{5 x + 4}{(x + 5) (x - 2)} ((x - 2) (x + 5))$

Step 3.2.1.6

Cancel the common factor of $(x - 2) (x + 5)$ .

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Step 3.2.1.6.1

Factor $(x - 2) (x + 5)$ out of $(x + 5) (x - 2)$ .

$r x + t = \frac{5 x + 4}{(x - 2) (x + 5) (1)} ((x - 2) (x + 5))$

Step 3.2.1.6.2

Cancel the common factor.

$r x + t = \frac{5 x + 4}{(x - 2) (x + 5) \cdot 1} ((x - 2) (x + 5))$

Step 3.2.1.6.3

Rewrite the expression.

$r x + t = 5 x + 4$

Step 4

Solve for $r$ .

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Step 4.1

Subtract $t$ from both sides of the equation.

$r x = 5 x + 4 - t$

Step 4.2

Divide each term in $r x = 5 x + 4 - t$ by $x$ and simplify.

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Step 4.2.1

Divide each term in $r x = 5 x + 4 - t$ by $x$ .

$\frac{r x}{x} = \frac{5 x}{x} + \frac{4}{x} + \frac{- t}{x}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $x$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{r x}{x} = \frac{5 x}{x} + \frac{4}{x} + \frac{- t}{x}$

Step 4.2.2.1.2

Divide $r$ by $1$ .

$r = \frac{5 x}{x} + \frac{4}{x} + \frac{- t}{x}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Cancel the common factor of $x$ .

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Step 4.2.3.1.1.1

Cancel the common factor.

$r = \frac{5 x}{x} + \frac{4}{x} + \frac{- t}{x}$

Step 4.2.3.1.1.2

Divide $5$ by $1$ .

$r = 5 + \frac{4}{x} + \frac{- t}{x}$

Step 4.2.3.1.2

Move the negative in front of the fraction.

$r = 5 + \frac{4}{x} - \frac{t}{x}$