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Equations Reducible to the Linear Form

3x+5/4x+2=3x+...

Question

$\frac{3 x + 5}{4 x + 2} = \frac{3 x + 4}{4 x + 7} .$ Solve this linear equation.

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Solution

Solving the equation:
We have, $\frac{3 x + 5}{4 x + 2} = \frac{3 x + 4}{4 x + 7}$
By cross-multiplying we get
$(3 x + 5) (4 x + 7) = (3 x + 4) (4 x + 2)$
Now applying the distributive property.
$\begin{array}{rcl} 3 x (4 x + 7) + 5 (4 x + 7) & = & 3 x (4 x + 2) + 4 (4 x + 2) \\ 12 x^{2} + 21 x + 20 x + 35 & = & 12 x^{2} + 6 x + 16 x + 8 \\ 41 x + 35 & = & 22 x + 8 \\ 19 x & = & - 27 \\ x & = & - \frac{27}{19} \end{array}$
Hence, the solution of the linear equation is $\begin{array}{rcl} - \frac{27}{19} \end{array}$

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