Question

Solve the following equations and verify your answer :

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

Answer 1

$\frac{3x+5}{4x+2}=\frac{3x+4}{4x+7}$
By cross multiplication :
$\Rightarrow (3x+5)(4x+7)=(3x+4)(4x+2)$
$\Rightarrow 12x^2+21x+20x+35=12x^2+6x+16x+8$
$\Rightarrow 12x^2+41x+35=12x^2+22x+8$
$\Rightarrow 12x^2+41x-12x^2-22x=8-35$
$\Rightarrow 19x=-27$

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

$\therefore x=\frac{-27}{19}$
Verification :
$L.H.S.=\frac{3x+5}{4x+2}$

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

$=\frac{\left(\frac{-81}{19}\right)+5}{\frac{(-108}{19})+2}$

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

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

$=\frac{14}{19}\times \frac{19}{-70}$

$=\frac{-14}{70}$

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

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

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

$=\frac{\left(\frac{-81}{19}\right)+4}{\left(\frac{-108}{19}\right)+7}$

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

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

$=\frac{-5}{19}\times \frac{19}{25}$
$=\frac{-1}{5}$
$\therefore L.H.S.=R.H.S.$

Hence, verified.