Question
Solve the following equation and verify your answer:
$\frac{3\text{x}+5}{4\text{x}+2}=\frac{3\text{x}+4}{4\text{x}+7}$
$\frac{3\text{x}+5}{4\text{x}+2}=\frac{3\text{x}+4}{4\text{x}+7}$
✓
Answer
$\frac{3\text{x}+5}{4\text{x}+2}=\frac{3\text{x}+4}{4\text{x}+7}$
By cross multiplication:
$(3\text{x}+5)(4\text{x}+7)=(3\text{x}+4)(4\text{x}+2)$
$\Rightarrow12\text{x}^2+21\text{x}+20\text{x}+35=12\text{x}^2+6\text{x}+16\text{x}+8$
$\Rightarrow12\text{x}^2+41\text{x}-12\text{x}^2-22\text{x}=8-35$
$\Rightarrow19\text{x}=-27$
$\Rightarrow\text{x}=\frac{-27}{19}$
$\therefore\text{x}=\frac{-27}{19}$
Verification:
$\text{L.H.S.}=\frac{3\text{x}+5}{4\text{x}+2}=\frac{3\Big(\frac{-27}{19}\Big)+5}{4\Big(\frac{-27}{19}\Big)+2}$
$=\frac{\frac{-81}{19}+5}{\frac{-108}{19}+2}=\frac{\frac{-81+95}{19}}{\frac{-108+38}{19}}=\frac{\frac{14}{19}}{\frac{-70}{19}}$
$\text{R.H.S.}=\frac{3\text{x}+4}{4\text{x}+7}$
$=\frac{3\Big(\frac{-27}{19}\Big)+4}{4\Big(\frac{-27}{19}\Big)+7}=\frac{\frac{-81}{19}+4}{\frac{-108}{19}+7}$
$=\frac{\frac{-81+76}{19}}{\frac{-108+133}{19}}=\frac{\frac{-5}{19}}{\frac{25}{19}}=\frac{-5}{19}\times\frac{19}{25}$
$=\frac{-1}{5}$
$\therefore\text{L.H.S.}=\text{R.H.S.}$
By cross multiplication:
$(3\text{x}+5)(4\text{x}+7)=(3\text{x}+4)(4\text{x}+2)$
$\Rightarrow12\text{x}^2+21\text{x}+20\text{x}+35=12\text{x}^2+6\text{x}+16\text{x}+8$
$\Rightarrow12\text{x}^2+41\text{x}-12\text{x}^2-22\text{x}=8-35$
$\Rightarrow19\text{x}=-27$
$\Rightarrow\text{x}=\frac{-27}{19}$
$\therefore\text{x}=\frac{-27}{19}$
Verification:
$\text{L.H.S.}=\frac{3\text{x}+5}{4\text{x}+2}=\frac{3\Big(\frac{-27}{19}\Big)+5}{4\Big(\frac{-27}{19}\Big)+2}$
$=\frac{\frac{-81}{19}+5}{\frac{-108}{19}+2}=\frac{\frac{-81+95}{19}}{\frac{-108+38}{19}}=\frac{\frac{14}{19}}{\frac{-70}{19}}$
$\text{R.H.S.}=\frac{3\text{x}+4}{4\text{x}+7}$
$=\frac{3\Big(\frac{-27}{19}\Big)+4}{4\Big(\frac{-27}{19}\Big)+7}=\frac{\frac{-81}{19}+4}{\frac{-108}{19}+7}$
$=\frac{\frac{-81+76}{19}}{\frac{-108+133}{19}}=\frac{\frac{-5}{19}}{\frac{25}{19}}=\frac{-5}{19}\times\frac{19}{25}$
$=\frac{-1}{5}$
$\therefore\text{L.H.S.}=\text{R.H.S.}$
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