Question

Solve the following equation and verify your answer:

xii) $\frac{3x+5}{4x+2}=\frac{3x+4}{4x+7}$

Answer 1

Given:

$\frac{3x+5}{4x+2}=\frac{3x+4}{4x+7}$

$\Rightarrow \frac{3x+5}{4x+2}-\frac{3x+4}{4x+7}=0$

$\Rightarrow \frac{(3x+5)(4x+7)–(3x+4)(4x+2)}{(4x+2)(4x+7)}=0$

$\Rightarrow (3x+5)(4x+7)–(3x+4)(4x+2)=0$

$\Rightarrow 12x^2+21x+20x+35–12x^2–6x–16x–8=0$

$\Rightarrow 19x+35–8=0$

$\Rightarrow 19x=-27$

$\Rightarrow x=-\frac{27}{19}$

To check: $\frac{3x+5}{4x+2}=\frac{3x+4}{4x+7}$ for $x=-\frac{27}{19}$

$L.H.S=\frac{3x+5}{4x+2}$

$=\frac{3(-\frac{27}{19})+5}{4\left(\frac{-27}{19}\right)+2}$

$=\frac{\frac{-81+95}{19}}{\frac{-108+38}{19}}$

$=\frac{\frac{14}{19}}{\frac{-70}{19}}$$=\frac{14}{-70}$$=-\frac{1}{5}$

$R.H.S=\frac{3x+4}{4x+7}$

$=\frac{3(-\frac{27}{19})+4}{4(-\frac{27}{19})+7}$

$=\frac{\frac{-81+76}{19}}{\frac{-108+133}{19}}$

$=\frac{-\frac{5}{19}}{\frac{25}{19}}$

$=-\frac{5}{25}$

$=-\frac{1}{5}$

$\therefore L.H.S=R.H.S$

Hence, the given equation is verified for $x=-\frac{27}{19}$.