Precalculus Examples

$\frac{7 x + 6}{(x + 6) (6 x - 1)}$

Step 1

Decompose the fraction and multiply through by the common denominator.

Tap for more steps...

Step 1.1

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $A$ .

$\frac{A}{x + 6}$

Step 1.2

$\frac{A}{x + 6} + \frac{B}{6 x - 1}$

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x + 6) (6 x - 1)$ .

$\frac{(7 x + 6) (x + 6) (6 x - 1)}{(x + 6) (6 x - 1)} = \frac{(A) (x + 6) (6 x - 1)}{x + 6} + \frac{(B) (x + 6) (6 x - 1)}{6 x - 1}$

Step 1.4

Cancel the common factor of $x + 6$ .

Tap for more steps...

Step 1.4.1

Cancel the common factor.

$\frac{(7 x + 6) (x + 6) (6 x - 1)}{(x + 6) (6 x - 1)} = \frac{(A) (x + 6) (6 x - 1)}{x + 6} + \frac{(B) (x + 6) (6 x - 1)}{6 x - 1}$

Step 1.4.2

Rewrite the expression.

$\frac{(7 x + 6) (6 x - 1)}{6 x - 1} = \frac{(A) (x + 6) (6 x - 1)}{x + 6} + \frac{(B) (x + 6) (6 x - 1)}{6 x - 1}$

Step 1.5

Cancel the common factor of $6 x - 1$ .

Tap for more steps...

Step 1.5.1

Cancel the common factor.

$\frac{(7 x + 6) (6 x - 1)}{6 x - 1} = \frac{(A) (x + 6) (6 x - 1)}{x + 6} + \frac{(B) (x + 6) (6 x - 1)}{6 x - 1}$

Step 1.5.2

Divide $7 x + 6$ by $1$ .

$7 x + 6 = \frac{(A) (x + 6) (6 x - 1)}{x + 6} + \frac{(B) (x + 6) (6 x - 1)}{6 x - 1}$

Step 1.6

Simplify each term.

Tap for more steps...

Step 1.6.1

Cancel the common factor of $x + 6$ .

Tap for more steps...

Step 1.6.1.1

Cancel the common factor.

$7 x + 6 = \frac{A (x + 6) (6 x - 1)}{x + 6} + \frac{(B) (x + 6) (6 x - 1)}{6 x - 1}$

Step 1.6.1.2

Divide $(A) (6 x - 1)$ by $1$ .

$7 x + 6 = (A) (6 x - 1) + \frac{(B) (x + 6) (6 x - 1)}{6 x - 1}$

Step 1.6.2

Apply the distributive property.

$7 x + 6 = A (6 x) + A \cdot - 1 + \frac{(B) (x + 6) (6 x - 1)}{6 x - 1}$

Step 1.6.3

Rewrite using the commutative property of multiplication.

$7 x + 6 = 6 A x + A \cdot - 1 + \frac{(B) (x + 6) (6 x - 1)}{6 x - 1}$

Step 1.6.4

Move $- 1$ to the left of $A$ .

$7 x + 6 = 6 A x - 1 \cdot A + \frac{(B) (x + 6) (6 x - 1)}{6 x - 1}$

Step 1.6.5

Rewrite $- 1 A$ as $- A$ .

$7 x + 6 = 6 A x - A + \frac{(B) (x + 6) (6 x - 1)}{6 x - 1}$

Step 1.6.6

Cancel the common factor of $6 x - 1$ .

Tap for more steps...

Step 1.6.6.1

Cancel the common factor.

$7 x + 6 = 6 A x - A + \frac{(B) (x + 6) (6 x - 1)}{6 x - 1}$

Step 1.6.6.2

Divide $(B) (x + 6)$ by $1$ .

$7 x + 6 = 6 A x - A + (B) (x + 6)$

Step 1.6.7

Apply the distributive property.

$7 x + 6 = 6 A x - A + B x + B \cdot 6$

Step 1.6.8

Move $6$ to the left of $B$ .

$7 x + 6 = 6 A x - A + B x + 6 B$

Step 1.7

Simplify the expression.

Tap for more steps...

Step 1.7.1

Move $A$ .

$7 x + 6 = 6 x A - A + B x + 6 B$

Step 1.7.2

Reorder $B$ and $x$ .

$7 x + 6 = 6 x A - A + B x + 6 B$

Step 1.7.3

Move $- A$ .

$7 x + 6 = 6 x A + B x - A + 6 B$

Step 2

Create equations for the partial fraction variables and use them to set up a system of equations.

Tap for more steps...

Step 2.1

Create an equation for the partial fraction variables by equating the coefficients of $x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$7 = 6 A + B$

Step 2.2

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$6 = - 1 A + 6 B$

Step 2.3

Set up the system of equations to find the coefficients of the partial fractions.

$7 = 6 A + B$

$6 = - 1 A + 6 B$

$7 = 6 A + B$

$6 = - 1 A + 6 B$

Step 3

Solve the system of equations.

Tap for more steps...

Step 3.1

Solve for $B$ in $7 = 6 A + B$ .

Tap for more steps...

Step 3.1.1

Rewrite the equation as $6 A + B = 7$ .

$6 A + B = 7$

$6 = - 1 A + 6 B$

Step 3.1.2

Subtract $6 A$ from both sides of the equation.

$B = 7 - 6 A$

$6 = - 1 A + 6 B$

$B = 7 - 6 A$

$6 = - 1 A + 6 B$

Step 3.2

Replace all occurrences of $B$ with $7 - 6 A$ in each equation.

Tap for more steps...

Step 3.2.1

Replace all occurrences of $B$ in $6 = - 1 A + 6 B$ with $7 - 6 A$ .

$6 = - 1 A + 6 (7 - 6 A)$

$B = 7 - 6 A$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Simplify $- 1 A + 6 (7 - 6 A)$ .

Tap for more steps...

Step 3.2.2.1.1

Simplify each term.

Tap for more steps...

Step 3.2.2.1.1.1

Rewrite $- 1 A$ as $- A$ .

$6 = - A + 6 (7 - 6 A)$

$B = 7 - 6 A$

Step 3.2.2.1.1.2

Apply the distributive property.

$6 = - A + 6 \cdot 7 + 6 (- 6 A)$

$B = 7 - 6 A$

Step 3.2.2.1.1.3

Multiply $6$ by $7$ .

$6 = - A + 42 + 6 (- 6 A)$

$B = 7 - 6 A$

Step 3.2.2.1.1.4

Multiply $- 6$ by $6$ .

$6 = - A + 42 - 36 A$

$B = 7 - 6 A$

$6 = - A + 42 - 36 A$

$B = 7 - 6 A$

Step 3.2.2.1.2

Subtract $36 A$ from $- A$ .

$6 = - 37 A + 42$

$B = 7 - 6 A$

$6 = - 37 A + 42$

$B = 7 - 6 A$

$6 = - 37 A + 42$

$B = 7 - 6 A$

$6 = - 37 A + 42$

$B = 7 - 6 A$

Step 3.3

Solve for $A$ in $6 = - 37 A + 42$ .

Tap for more steps...

Step 3.3.1

Rewrite the equation as $- 37 A + 42 = 6$ .

$- 37 A + 42 = 6$

$B = 7 - 6 A$

Step 3.3.2

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

Tap for more steps...

Step 3.3.2.1

Subtract $42$ from both sides of the equation.

$- 37 A = 6 - 42$

$B = 7 - 6 A$

Step 3.3.2.2

Subtract $42$ from $6$ .

$- 37 A = - 36$

$B = 7 - 6 A$

$- 37 A = - 36$

$B = 7 - 6 A$

Step 3.3.3

Divide each term in $- 37 A = - 36$ by $- 37$ and simplify.

Tap for more steps...

Step 3.3.3.1

Divide each term in $- 37 A = - 36$ by $- 37$ .

$\frac{- 37 A}{- 37} = \frac{- 36}{- 37}$

$B = 7 - 6 A$

Step 3.3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.3.2.1

Cancel the common factor of $- 37$ .

Tap for more steps...

Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 37 A}{- 37} = \frac{- 36}{- 37}$

$B = 7 - 6 A$

Step 3.3.3.2.1.2

Divide $A$ by $1$ .

$A = \frac{- 36}{- 37}$

$B = 7 - 6 A$

$A = \frac{- 36}{- 37}$

$B = 7 - 6 A$

$A = \frac{- 36}{- 37}$

$B = 7 - 6 A$

Step 3.3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.3.1

Dividing two negative values results in a positive value.

$A = \frac{36}{37}$

$B = 7 - 6 A$

$A = \frac{36}{37}$

$B = 7 - 6 A$

$A = \frac{36}{37}$

$B = 7 - 6 A$

$A = \frac{36}{37}$

$B = 7 - 6 A$

Step 3.4

Replace all occurrences of $A$ with $\frac{36}{37}$ in each equation.

Tap for more steps...

Step 3.4.1

Replace all occurrences of $A$ in $B = 7 - 6 A$ with $\frac{36}{37}$ .

$B = 7 - 6 (\frac{36}{37})$

$A = \frac{36}{37}$

Step 3.4.2

Simplify the right side.

Tap for more steps...

Step 3.4.2.1

Simplify $7 - 6 (\frac{36}{37})$ .

Tap for more steps...

Step 3.4.2.1.1

Simplify each term.

Tap for more steps...

Step 3.4.2.1.1.1

Multiply $- 6 (\frac{36}{37})$ .

Tap for more steps...

Step 3.4.2.1.1.1.1

Combine $- 6$ and $\frac{36}{37}$ .

$B = 7 + \frac{- 6 \cdot 36}{37}$

$A = \frac{36}{37}$

Step 3.4.2.1.1.1.2

Multiply $- 6$ by $36$ .

$B = 7 + \frac{- 216}{37}$

$A = \frac{36}{37}$

$B = 7 + \frac{- 216}{37}$

$A = \frac{36}{37}$

Step 3.4.2.1.1.2

Move the negative in front of the fraction.

$B = 7 - \frac{216}{37}$

$A = \frac{36}{37}$

$B = 7 - \frac{216}{37}$

$A = \frac{36}{37}$

Step 3.4.2.1.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{37}{37}$ .

$B = 7 \cdot \frac{37}{37} - \frac{216}{37}$

$A = \frac{36}{37}$

Step 3.4.2.1.3

Combine $7$ and $\frac{37}{37}$ .

$B = \frac{7 \cdot 37}{37} - \frac{216}{37}$

$A = \frac{36}{37}$

Step 3.4.2.1.4

Combine the numerators over the common denominator.

$B = \frac{7 \cdot 37 - 216}{37}$

$A = \frac{36}{37}$

Step 3.4.2.1.5

Simplify the numerator.

Tap for more steps...

Step 3.4.2.1.5.1

Multiply $7$ by $37$ .

$B = \frac{259 - 216}{37}$

$A = \frac{36}{37}$

Step 3.4.2.1.5.2

Subtract $216$ from $259$ .

$B = \frac{43}{37}$

$A = \frac{36}{37}$

$B = \frac{43}{37}$

$A = \frac{36}{37}$

$B = \frac{43}{37}$

$A = \frac{36}{37}$

$B = \frac{43}{37}$

$A = \frac{36}{37}$

$B = \frac{43}{37}$

$A = \frac{36}{37}$

Step 3.5

List all of the solutions.

$B = \frac{43}{37}, A = \frac{36}{37}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x + 6} + \frac{B}{6 x - 1}$ with the values found for $A$ and $B$ .

$\frac{\frac{36}{37}}{x + 6} + \frac{\frac{43}{37}}{6 x - 1}$