Solve by Factoring a/(a-6)=-5/(a-4)

$\frac{a}{a - 6} = \frac{- 5}{a - 4}$

Step 1

Subtract $\frac{- 5}{a - 4}$ from both sides of the equation.

$\frac{a}{a - 6} - \frac{- 5}{a - 4} = 0$

Step 2

Simplify $\frac{a}{a - 6} - \frac{- 5}{a - 4}$ .

Simplify each term.

Move the negative in front of the fraction.

$\frac{a}{a - 6} - - \frac{5}{a - 4} = 0$

Multiply $- - \frac{5}{a - 4}$ .

Multiply $- 1$ by $- 1$ .

$\frac{a}{a - 6} + 1 \frac{5}{a - 4} = 0$

Multiply $\frac{5}{a - 4}$ by $1$ .

$\frac{a}{a - 6} + \frac{5}{a - 4} = 0$

To write $\frac{a}{a - 6}$ as a fraction with a common denominator, multiply by $\frac{a - 4}{a - 4}$ .

$\frac{a}{a - 6} \cdot \frac{a - 4}{a - 4} + \frac{5}{a - 4} = 0$

To write $\frac{5}{a - 4}$ as a fraction with a common denominator, multiply by $\frac{a - 6}{a - 6}$ .

$\frac{a}{a - 6} \cdot \frac{a - 4}{a - 4} + \frac{5}{a - 4} \cdot \frac{a - 6}{a - 6} = 0$

Write each expression with a common denominator of $(a - 6) (a - 4)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{a}{a - 6}$ by $\frac{a - 4}{a - 4}$ .

$\frac{a (a - 4)}{(a - 6) (a - 4)} + \frac{5}{a - 4} \cdot \frac{a - 6}{a - 6} = 0$

Multiply $\frac{5}{a - 4}$ by $\frac{a - 6}{a - 6}$ .

$\frac{a (a - 4)}{(a - 6) (a - 4)} + \frac{5 (a - 6)}{(a - 4) (a - 6)} = 0$

Reorder the factors of $(a - 6) (a - 4)$ .

$\frac{a (a - 4)}{(a - 4) (a - 6)} + \frac{5 (a - 6)}{(a - 4) (a - 6)} = 0$

Combine the numerators over the common denominator.

$\frac{a (a - 4) + 5 (a - 6)}{(a - 4) (a - 6)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{a \cdot a + a \cdot - 4 + 5 (a - 6)}{(a - 4) (a - 6)} = 0$

Multiply $a$ by $a$ .

$\frac{a^{2} + a \cdot - 4 + 5 (a - 6)}{(a - 4) (a - 6)} = 0$

Move $- 4$ to the left of $a$ .

$\frac{a^{2} - 4 \cdot a + 5 (a - 6)}{(a - 4) (a - 6)} = 0$

Apply the distributive property.

$\frac{a^{2} - 4 a + 5 a + 5 \cdot - 6}{(a - 4) (a - 6)} = 0$

Multiply $5$ by $- 6$ .

$\frac{a^{2} - 4 a + 5 a - 30}{(a - 4) (a - 6)} = 0$

Add $- 4 a$ and $5 a$ .

$\frac{a^{2} + a - 30}{(a - 4) (a - 6)} = 0$

Factor $a^{2} + a - 30$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 30$ and whose sum is $1$ .

$- 5, 6$

Write the factored form using these integers.

$\frac{(a - 5) (a + 6)}{(a - 4) (a - 6)} = 0$

Step 3

Set the numerator equal to zero.

$(a - 5) (a + 6) = 0$

Step 4

Solve the equation for $a$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$a - 5 = 0$

$a + 6 = 0$

Set $a - 5$ equal to $0$ and solve for $a$ .

Set $a - 5$ equal to $0$ .

$a - 5 = 0$

Add $5$ to both sides of the equation.

$a = 5$

Set $a + 6$ equal to $0$ and solve for $a$ .

Set $a + 6$ equal to $0$ .

$a + 6 = 0$

Subtract $6$ from both sides of the equation.

$a = - 6$

The final solution is all the values that make $(a - 5) (a + 6) = 0$ true.

$a = 5, - 6$

Solve by Factoring a/(a-6)=-5/(a-4)

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