Algebra Examples

$\frac{2 x + 5}{(x - 8) (2 x + 1)} = \frac{a}{x - 8} + \frac{b}{2 x + 1}$

Step 1

Rewrite the equation as $\frac{a}{x - 8} + \frac{b}{2 x + 1} = \frac{2 x + 5}{(x - 8) (2 x + 1)}$ .

$\frac{a}{x - 8} + \frac{b}{2 x + 1} = \frac{2 x + 5}{(x - 8) (2 x + 1)}$

Step 2

Move all terms containing variables to the left side of the equation.

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Step 2.1

Subtract $\frac{2 x + 5}{(x - 8) (2 x + 1)}$ from both sides of the equation.

$\frac{a}{x - 8} + \frac{b}{2 x + 1} - \frac{2 x + 5}{(x - 8) (2 x + 1)} = 0$

Step 2.2

To write $\frac{a}{x - 8}$ as a fraction with a common denominator, multiply by $\frac{2 x + 1}{2 x + 1}$ .

$\frac{a}{x - 8} \cdot \frac{2 x + 1}{2 x + 1} + \frac{b}{2 x + 1} - \frac{2 x + 5}{(x - 8) (2 x + 1)} = 0$

Step 2.3

To write $\frac{b}{2 x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 8}{x - 8}$ .

$\frac{a}{x - 8} \cdot \frac{2 x + 1}{2 x + 1} + \frac{b}{2 x + 1} \cdot \frac{x - 8}{x - 8} - \frac{2 x + 5}{(x - 8) (2 x + 1)} = 0$

Step 2.4

Write each expression with a common denominator of $(x - 8) (2 x + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{a}{x - 8}$ by $\frac{2 x + 1}{2 x + 1}$ .

$\frac{a (2 x + 1)}{(x - 8) (2 x + 1)} + \frac{b}{2 x + 1} \cdot \frac{x - 8}{x - 8} - \frac{2 x + 5}{(x - 8) (2 x + 1)} = 0$

Step 2.4.2

Multiply $\frac{b}{2 x + 1}$ by $\frac{x - 8}{x - 8}$ .

$\frac{a (2 x + 1)}{(x - 8) (2 x + 1)} + \frac{b (x - 8)}{(2 x + 1) (x - 8)} - \frac{2 x + 5}{(x - 8) (2 x + 1)} = 0$

Step 2.4.3

Reorder the factors of $(x - 8) (2 x + 1)$ .

$\frac{a (2 x + 1)}{(2 x + 1) (x - 8)} + \frac{b (x - 8)}{(2 x + 1) (x - 8)} - \frac{2 x + 5}{(x - 8) (2 x + 1)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{a (2 x + 1) + b (x - 8)}{(2 x + 1) (x - 8)} - \frac{2 x + 5}{(x - 8) (2 x + 1)} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Apply the distributive property.

$\frac{a (2 x) + a \cdot 1 + b (x - 8)}{(2 x + 1) (x - 8)} - \frac{2 x + 5}{(x - 8) (2 x + 1)} = 0$

Step 2.6.2

Rewrite using the commutative property of multiplication.

$\frac{2 a x + a \cdot 1 + b (x - 8)}{(2 x + 1) (x - 8)} - \frac{2 x + 5}{(x - 8) (2 x + 1)} = 0$

Step 2.6.3

Multiply $a$ by $1$ .

$\frac{2 a x + a + b (x - 8)}{(2 x + 1) (x - 8)} - \frac{2 x + 5}{(x - 8) (2 x + 1)} = 0$

Step 2.6.4

Apply the distributive property.

$\frac{2 a x + a + b x + b \cdot - 8}{(2 x + 1) (x - 8)} - \frac{2 x + 5}{(x - 8) (2 x + 1)} = 0$

Step 2.6.5

Move $- 8$ to the left of $b$ .

$\frac{2 a x + a + b x - 8 b}{(2 x + 1) (x - 8)} - \frac{2 x + 5}{(x - 8) (2 x + 1)} = 0$

Step 2.7

Reorder the factors of $(x - 8) (2 x + 1)$ .

$\frac{2 a x + a + b x - 8 b}{(2 x + 1) (x - 8)} - \frac{2 x + 5}{(2 x + 1) (x - 8)} = 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{2 a x + a + b x - 8 b - (2 x + 5)}{(2 x + 1) (x - 8)} = 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Apply the distributive property.

$\frac{2 a x + a + b x - 8 b - (2 x) - 1 \cdot 5}{(2 x + 1) (x - 8)} = 0$

Step 2.9.2

Multiply $2$ by $- 1$ .

$\frac{2 a x + a + b x - 8 b - 2 x - 1 \cdot 5}{(2 x + 1) (x - 8)} = 0$

Step 2.9.3

Multiply $- 1$ by $5$ .

$\frac{2 a x + a + b x - 8 b - 2 x - 5}{(2 x + 1) (x - 8)} = 0$

Step 3

Set the numerator equal to zero.

$2 a x + a + b x - 8 b - 2 x - 5 = 0$

Step 4

Solve the equation for $a$ .

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Step 4.1

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

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Step 4.1.1

Subtract $b x$ from both sides of the equation.

$2 a x + a - 8 b - 2 x - 5 = - b x$

Step 4.1.2

Add $8 b$ to both sides of the equation.

$2 a x + a - 2 x - 5 = - b x + 8 b$

Step 4.1.3

Add $2 x$ to both sides of the equation.

$2 a x + a - 5 = - b x + 8 b + 2 x$

Step 4.1.4

Add $5$ to both sides of the equation.

$2 a x + a = - b x + 8 b + 2 x + 5$

Step 4.2

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

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Step 4.2.1

Factor $a$ out of $2 a x$ .

$a (2 x) + a = - b x + 8 b + 2 x + 5$

Step 4.2.2

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

$a (2 x) + a = - b x + 8 b + 2 x + 5$

Step 4.2.3

Factor $a$ out of $a^{1}$ .

$a (2 x) + a \cdot 1 = - b x + 8 b + 2 x + 5$

Step 4.2.4

Factor $a$ out of $a (2 x) + a \cdot 1$ .

$a (2 x + 1) = - b x + 8 b + 2 x + 5$

Step 4.3

Divide each term in $a (2 x + 1) = - b x + 8 b + 2 x + 5$ by $2 x + 1$ and simplify.

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Step 4.3.1

Divide each term in $a (2 x + 1) = - b x + 8 b + 2 x + 5$ by $2 x + 1$ .

$\frac{a (2 x + 1)}{2 x + 1} = \frac{- b x}{2 x + 1} + \frac{8 b}{2 x + 1} + \frac{2 x}{2 x + 1} + \frac{5}{2 x + 1}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $2 x + 1$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{a (2 x + 1)}{2 x + 1} = \frac{- b x}{2 x + 1} + \frac{8 b}{2 x + 1} + \frac{2 x}{2 x + 1} + \frac{5}{2 x + 1}$

Step 4.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- b x}{2 x + 1} + \frac{8 b}{2 x + 1} + \frac{2 x}{2 x + 1} + \frac{5}{2 x + 1}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Combine into one fraction.

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Step 4.3.3.1.1

Move the negative in front of the fraction.

$a = - \frac{b x}{2 x + 1} + \frac{8 b}{2 x + 1} + \frac{2 x}{2 x + 1} + \frac{5}{2 x + 1}$

Step 4.3.3.1.2

Combine the numerators over the common denominator.

$a = \frac{- b x + 8 b}{2 x + 1} + \frac{2 x}{2 x + 1} + \frac{5}{2 x + 1}$

Step 4.3.3.2

Simplify the numerator.

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Step 4.3.3.2.1

Factor $b$ out of $- b x + 8 b$ .

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Step 4.3.3.2.1.1

Factor $b$ out of $- b x$ .

$a = \frac{b (- 1 x) + 8 b}{2 x + 1} + \frac{2 x}{2 x + 1} + \frac{5}{2 x + 1}$

Step 4.3.3.2.1.2

Factor $b$ out of $8 b$ .

$a = \frac{b (- 1 x) + b \cdot 8}{2 x + 1} + \frac{2 x}{2 x + 1} + \frac{5}{2 x + 1}$

Step 4.3.3.2.1.3

Factor $b$ out of $b (- 1 x) + b \cdot 8$ .

$a = \frac{b (- 1 x + 8)}{2 x + 1} + \frac{2 x}{2 x + 1} + \frac{5}{2 x + 1}$

Step 4.3.3.2.2

Rewrite $- 1 x$ as $- x$ .

$a = \frac{b (- x + 8)}{2 x + 1} + \frac{2 x}{2 x + 1} + \frac{5}{2 x + 1}$

Step 4.3.3.3

Combine the numerators over the common denominator.

$a = \frac{b (- x + 8) + 2 x}{2 x + 1} + \frac{5}{2 x + 1}$

Step 4.3.3.4

Simplify the numerator.

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Step 4.3.3.4.1

Apply the distributive property.

$a = \frac{b (- x) + b \cdot 8 + 2 x}{2 x + 1} + \frac{5}{2 x + 1}$

Step 4.3.3.4.2

Rewrite using the commutative property of multiplication.

$a = \frac{- b x + b \cdot 8 + 2 x}{2 x + 1} + \frac{5}{2 x + 1}$

Step 4.3.3.4.3

Move $8$ to the left of $b$ .

$a = \frac{- b x + 8 b + 2 x}{2 x + 1} + \frac{5}{2 x + 1}$

Step 4.3.3.5

Simplify terms.

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Step 4.3.3.5.1

Combine the numerators over the common denominator.

$a = \frac{- b x + 8 b + 2 x + 5}{2 x + 1}$

Step 4.3.3.5.2

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

$a = \frac{- (b x) + 8 b + 2 x + 5}{2 x + 1}$

Step 4.3.3.5.3

Factor $- 1$ out of $8 b$ .

$a = \frac{- (b x) - (- 8 b) + 2 x + 5}{2 x + 1}$

Step 4.3.3.5.4

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

$a = \frac{- (b x - 8 b) + 2 x + 5}{2 x + 1}$

Step 4.3.3.5.5

Factor $- 1$ out of $2 x$ .

$a = \frac{- (b x - 8 b) - (- 2 x) + 5}{2 x + 1}$

Step 4.3.3.5.6

Factor $- 1$ out of $- (b x - 8 b) - (- 2 x)$ .

$a = \frac{- (b x - 8 b - 2 x) + 5}{2 x + 1}$

Step 4.3.3.5.7

Rewrite $5$ as $- 1 (- 5)$ .

$a = \frac{- (b x - 8 b - 2 x) - 1 (- 5)}{2 x + 1}$

Step 4.3.3.5.8

Factor $- 1$ out of $- (b x - 8 b - 2 x) - 1 (- 5)$ .

$a = \frac{- (b x - 8 b - 2 x - 5)}{2 x + 1}$

Step 4.3.3.5.9

Simplify the expression.

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Step 4.3.3.5.9.1

Rewrite $- (b x - 8 b - 2 x - 5)$ as $- 1 (b x - 8 b - 2 x - 5)$ .

$a = \frac{- 1 (b x - 8 b - 2 x - 5)}{2 x + 1}$

Step 4.3.3.5.9.2

Move the negative in front of the fraction.

$a = - \frac{b x - 8 b - 2 x - 5}{2 x + 1}$