# Solve the following rational equations. a) \frac{1}{3a + 18} = 1 - \frac{8}{a + 6} b)...

## Question:

Solve the following rational equations.

a) {eq}\displaystyle\;\frac{1}{3a + 18} = 1 - \frac{8}{a + 6} {/eq}

b) {eq}\displaystyle\;\frac{7}{2x^2 - x - 1} = \frac{5}{x - 1} {/eq}

## Rational Equations:

There are two parts to this question. In one rational equation, there are linear terms and in the other, we have quadratic terms. We will apply the splitting the middle term technique to simplify the expression and solve the equation.

a)

$$\displaystyle\;\frac{1}{3a + 18} = 1 - \frac{8}{a + 6}\\ \frac{1}{3(a+6)}=1-\frac{8}{a+6}\\ \frac{1}{3(a+6)}+\frac{8}{a+6}=1$$

We will take the L.C.M.

$$\frac{25}{3(a+6)}=1\\ a+6=\frac{25}{3}\\ a=\frac{7}{3}$$

b)

$$\displaystyle\;\frac{7}{2x^2 - x - 1} = \frac{5}{x - 1}$$

We will factorise the quadratic term using the splitting the middle term technique:

$$\frac{7}{2x^2-2x+x-1}=\frac{5}{x-1}\\ \frac{7}{2x(x-1)+1(x-1)}=\frac{5}{x-1}\\ \frac{7}{(x-1)(2x+1)}=\frac{5}{x-1}$$

Cancelling the common term:

$$\frac{7}{2x+1}=5\\ 7=10x+5\\ x=\frac{1}{5}$$