Solve by Factoring a/(a-7)=-5/(a-3)

$\frac{a}{a - 7} = \frac{- 5}{a - 3}$

Step 1

Subtract $\frac{- 5}{a - 3}$ from both sides of the equation.

$\frac{a}{a - 7} - \frac{- 5}{a - 3} = 0$

Step 2

Simplify $\frac{a}{a - 7} - \frac{- 5}{a - 3}$ .

Simplify each term.

Move the negative in front of the fraction.

$\frac{a}{a - 7} - - \frac{5}{a - 3} = 0$

Multiply $- - \frac{5}{a - 3}$ .

Multiply $- 1$ by $- 1$ .

$\frac{a}{a - 7} + 1 \frac{5}{a - 3} = 0$

Multiply $\frac{5}{a - 3}$ by $1$ .

$\frac{a}{a - 7} + \frac{5}{a - 3} = 0$

To write $\frac{a}{a - 7}$ as a fraction with a common denominator, multiply by $\frac{a - 3}{a - 3}$ .

$\frac{a}{a - 7} \cdot \frac{a - 3}{a - 3} + \frac{5}{a - 3} = 0$

To write $\frac{5}{a - 3}$ as a fraction with a common denominator, multiply by $\frac{a - 7}{a - 7}$ .

$\frac{a}{a - 7} \cdot \frac{a - 3}{a - 3} + \frac{5}{a - 3} \cdot \frac{a - 7}{a - 7} = 0$

Write each expression with a common denominator of $(a - 7) (a - 3)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{a}{a - 7}$ by $\frac{a - 3}{a - 3}$ .

$\frac{a (a - 3)}{(a - 7) (a - 3)} + \frac{5}{a - 3} \cdot \frac{a - 7}{a - 7} = 0$

Multiply $\frac{5}{a - 3}$ by $\frac{a - 7}{a - 7}$ .

$\frac{a (a - 3)}{(a - 7) (a - 3)} + \frac{5 (a - 7)}{(a - 3) (a - 7)} = 0$

Reorder the factors of $(a - 7) (a - 3)$ .

$\frac{a (a - 3)}{(a - 3) (a - 7)} + \frac{5 (a - 7)}{(a - 3) (a - 7)} = 0$

Combine the numerators over the common denominator.

$\frac{a (a - 3) + 5 (a - 7)}{(a - 3) (a - 7)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{a \cdot a + a \cdot - 3 + 5 (a - 7)}{(a - 3) (a - 7)} = 0$

Multiply $a$ by $a$ .

$\frac{a^{2} + a \cdot - 3 + 5 (a - 7)}{(a - 3) (a - 7)} = 0$

Move $- 3$ to the left of $a$ .

$\frac{a^{2} - 3 \cdot a + 5 (a - 7)}{(a - 3) (a - 7)} = 0$

Apply the distributive property.

$\frac{a^{2} - 3 a + 5 a + 5 \cdot - 7}{(a - 3) (a - 7)} = 0$

Multiply $5$ by $- 7$ .

$\frac{a^{2} - 3 a + 5 a - 35}{(a - 3) (a - 7)} = 0$

Add $- 3 a$ and $5 a$ .

$\frac{a^{2} + 2 a - 35}{(a - 3) (a - 7)} = 0$

Factor $a^{2} + 2 a - 35$ 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 $- 35$ and whose sum is $2$ .

$- 5, 7$

Write the factored form using these integers.

$\frac{(a - 5) (a + 7)}{(a - 3) (a - 7)} = 0$

Step 3

Set the numerator equal to zero.

$(a - 5) (a + 7) = 0$

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 $0$ .

$a - 5 = 0$

$a + 7 = 0$

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 $a + 7$ equal to $0$ and solve for $a$ .

Set $a + 7$ equal to $0$ .

$a + 7 = 0$

Subtract $7$ from both sides of the equation.

$a = - 7$

The final solution is all the values that make $(a - 5) (a + 7) = 0$ true.

$a = 5, - 7$

Solve by Factoring a/(a-7)=-5/(a-3)

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