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Linear Equation in 2 Variables

Solve for x...

Question

Solve for $x, 2 (\frac{2 x + 3}{x - 3}) - 25 (\frac{x - 3}{2 x + 3}) = 5$

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Solution

We have,
$2 (\frac{2 x + 3}{x - 3}) - 25 (\frac{x - 3}{2 x + 3}) = 5$

$\Rightarrow$ $\frac{2 (2 x + 3)^{2} - 25 (x - 3)^{2}}{(x - 3) (2 x + 3)} = 5$

$\Rightarrow$ $2 (2 x + 3)^{2} - 25 (x - 3)^{2} = 5 (x - 3) (2 x + 3)$

$\Rightarrow$ $2 (4 x^{2} + 9 + 12 x) - 25 (x^{2} + 9 - 6 x) = 5 (2 x^{2} + 3 x - 6 x - 9)$

$\Rightarrow$ $8 x^{2} + 18 + 24 x - 25 x^{2} - 225 + 150 x = 10 x^{2} + 15 x - 30 x - 45$

$\Rightarrow$ $- 17 x^{2} + 174 x - 207 = 10 x^{2} - 15 x - 45$

$\Rightarrow$ $27 x^{2} - 189 x + 162 = 0$

$\Rightarrow$ $3 x^{2} - 21 x + 36 = 0$

$\Rightarrow$ $3 x^{2} - 21 x + 36 = 0$

$\Rightarrow$ $3 x^{2} - 9 x - 12 x + 36 = 0$

$\Rightarrow$ $3 x (x - 3) - 12 (x - 3) = 0$

$\Rightarrow$ $(x - 3) (3 x - 12) = 0$

$\Rightarrow x = 3; x = 4$

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