Solve for x and y:

$6 x + 5 y = 7 x + 3 y + 1 = 2 (x + 6 y - 1)$

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Solution

$6 x + 5 y = 7 x + 3 y + 1 = 2 (x + 6 y - 1)$

$6 x + 5 y = 7 x + 3 y + 1$

$\Rightarrow 7 x + 3 y + 1 = 6 x + 5 y$

$\Rightarrow 7 x - 6 x + 3 y - 5 y + 1 = 0$

$\Rightarrow x - 2 y + 1 = 0$

$\Rightarrow x = 2 y - 1$ ---(1)

$2 (x + 6 y - 1) = 6 x + 5 y$

$\Rightarrow 2 (2 y - 1 + 6 y - 1) = 6 (2 y - 1) + 5 y$

$\Rightarrow 2 (8 y - 2) = 12 y - 6 + 5 y$

$\Rightarrow 16 y - 4 = 17 y - 6$

$\Rightarrow 17 y - 16 y = 6 - 4$

$\Rightarrow y = 2$

Substitute this in (1),

$x = 2 (2) - 1 = 4 - 1 = 3$

So, x = 3 and y = 2

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6 x + 5 y = 7 x + 3 y + 1 = 2 (x + 6 y - 1)

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