# How do you solve the rational equation a/(a+5) = 3/(a+1)?

May 29, 2018

$a = - 3 , a = 5$

#### Explanation:

$\frac{a}{a + 5} = \frac{3}{a + 1}$

cross multiply:

$a \left(a + 1\right) = 3 \left(a + 5\right)$

${a}^{2} + a = 3 a + 15$

${a}^{2} - 2 a - 15 = 0$
$\left(a + 3\right) \left(a - 5\right) = 0$

$a = - 3 , a = 5$

May 29, 2018

$a = - 3 \text{ or } a = 5$

#### Explanation:

$\text{multiply through by } \left(a + 5\right) \left(a + 1\right)$

$\cancel{\left(a + 5\right)} \left(a + 1\right) \times \frac{a}{\cancel{a + 5}} = \left(a + 5\right) \cancel{\left(a + 1\right)} \times \frac{3}{\cancel{a + 1}}$

$a \left(a + 1\right) = 3 \left(a + 5\right) \leftarrow \textcolor{b l u e}{\text{distribute}}$

${a}^{2} + a = 3 a + 15$

"rearrange in "color(blue)"standard form ";ax^2+bx+c=0

$\text{subtract "3a+15" from both sides}$

${a}^{2} - 2 a - 15 = 0$

$\text{the factors of - 15 which sum to - 2 are - 5 and + 3}$

$\left(a - 5\right) \left(a + 3\right) = 0$

$\text{equate each factor to zero and solve for a}$

$a + 3 = 0 \Rightarrow a = - 3$

$a - 5 = 0 \Rightarrow a = 5$