Question
Solve the following equation and verify your answer: $\frac{9\text{x}-7}{3\text{x}+5}=\frac{3\text{x}-4}{\text{x}+6}$
✓
Answer
$\frac{9\text{x}-7}{3\text{x}+5}=\frac{3\text{x}-4}{\text{x}+6}$
By cross multiplication, $(9\text{x}-7)(\text{x}+6)=(3\text{x}-4)(3\text{x}+5)$
$\Rightarrow9\text{x}^2+54\text{x}-7\text{x}-42=9\text{x}^2+15\text{x}-12\text{x}-20$
$\Rightarrow9\text{x}^2+47\text{x}-42=9\text{x}^2+3\text{x}-20$
$\Rightarrow9\text{x}^2+47\text{x}-9\text{x}^2-3\text{x}=-20+42$ (By transposition)
$\Rightarrow44\text{x}$
$=22$
$\Rightarrow\text{x}=\frac{22}{44}$
$=\frac{1}{2}$
$\therefore\text{x}=\frac{1}{2}$
Verification: $\text{L.H.S}=\frac{9\text{x}-7}{3\text{x}+5}=\frac{9\times\frac{1}{2}-7}{3\times\frac{1}{2}+5}=\frac{\frac{9}{2}-7}{\frac{3}{2}+5}$
$=\frac{\frac{9-14}{2}}{\frac{3+10}{2}}=\frac{\frac{-5}{2}}{\frac{13}{2}}=\frac{-5}{2}\times\frac{2}{13}=\frac{-5}{13}$
$\text{R.H.S}=\frac{3\text{x}-4}{\text{x+6}}=\frac{3\times\frac{1}{2}-4}{\frac{1}{2}+6}=\frac{\frac{3}{2}-4}{\frac{1}{2}+6}$
$=\frac{\frac{3-8}{2}}{\frac{1+12}{2}}=\frac{\frac{-5}{2}}{\frac{13}{2}}=\frac{-5}{2}\times\frac{2}{13}=\frac{-5}{13}$
$\therefore\text{L.H.S}=\text{R.H.S}$
By cross multiplication, $(9\text{x}-7)(\text{x}+6)=(3\text{x}-4)(3\text{x}+5)$
$\Rightarrow9\text{x}^2+54\text{x}-7\text{x}-42=9\text{x}^2+15\text{x}-12\text{x}-20$
$\Rightarrow9\text{x}^2+47\text{x}-42=9\text{x}^2+3\text{x}-20$
$\Rightarrow9\text{x}^2+47\text{x}-9\text{x}^2-3\text{x}=-20+42$ (By transposition)
$\Rightarrow44\text{x}$
$=22$
$\Rightarrow\text{x}=\frac{22}{44}$
$=\frac{1}{2}$
$\therefore\text{x}=\frac{1}{2}$
Verification: $\text{L.H.S}=\frac{9\text{x}-7}{3\text{x}+5}=\frac{9\times\frac{1}{2}-7}{3\times\frac{1}{2}+5}=\frac{\frac{9}{2}-7}{\frac{3}{2}+5}$
$=\frac{\frac{9-14}{2}}{\frac{3+10}{2}}=\frac{\frac{-5}{2}}{\frac{13}{2}}=\frac{-5}{2}\times\frac{2}{13}=\frac{-5}{13}$
$\text{R.H.S}=\frac{3\text{x}-4}{\text{x+6}}=\frac{3\times\frac{1}{2}-4}{\frac{1}{2}+6}=\frac{\frac{3}{2}-4}{\frac{1}{2}+6}$
$=\frac{\frac{3-8}{2}}{\frac{1+12}{2}}=\frac{\frac{-5}{2}}{\frac{13}{2}}=\frac{-5}{2}\times\frac{2}{13}=\frac{-5}{13}$
$\therefore\text{L.H.S}=\text{R.H.S}$
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