Solve $\frac{9 x - 7}{3 x + 5} = \frac{3 x - 4}{x + 6}$ and choose the option which represents the solution correctly.

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$x = 2$

$x = - \frac{1}{2}$

$x = \frac{1}{2}$

$x = 1$

answer is A.

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Detailed Solution

The given linear equation is as follows,
$\frac{9 x - 7}{3 x + 5} = \frac{3 x - 4}{x + 6}$ .
We first cross-multiply the denominators with the numerators on the either side as follows and simplify.
$\begin{matrix} (9 x - 7) (x + 6) = (3 x - 4) (3 x + 5) \\ 9 x (x + 6) - 7 (x + 6) = 3 x (3 x + 5) - 4 (3 x + 5) \\ 9 x^{2} + 54 x - 7 x - 42 = 9 x^{2} + 15 x - 12 x - 20 \\ 9 x^{2} + 47 x - 42 = 9 x^{2} + 3 x - 20 \end{matrix}$
$\begin{matrix} 9 x^{2} + 47 x - 42 - 9 x^{2} = 9 x^{2} + 3 x - 20 - 9 x^{2} \\ 47 x - 42 = 3 x - 20 \\ 47 x = 3 x + 22 \\ 47 x - 3 x = 3 x - 3 x + 22 \\ 44 x = 22 \end{matrix}$
$\begin{matrix} \frac{44 x}{44} = \frac{22}{44} \\ x = \frac{1}{2} \end{matrix}$
So, the value of the variable x is $\frac{1}{2}$ .