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Variable on Both Sides

Solve for x a...

Question

Solve for $x a n d y$
$\frac{3 x - 2}{3 y + 7} = \frac{5 x - 1}{5 y + 16}; \frac{3 x - 15}{x - 9} = \frac{6 y - 5}{2 y + 3}$

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Solution

Step 1:Solving the equations by cross multiplication method
Given,
$\frac{3 x - 2}{3 y + 7} = \frac{5 x - 1}{5 y + 16} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) \frac{3 x - 15}{x - 9} = \frac{6 y - 5}{2 y + 3} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)$
We have,
$\frac{3 x - 2}{3 y + 7} = \frac{5 x - 1}{5 y + 16} \Rightarrow (3 x - 2) (5 y + 16) = (3 y + 7) (5 x - 1) \Rightarrow 15 x y + 48 x - 10 y - 32 = 15 x y - 3 y + 35 x - 7 \Rightarrow 15 x y + 48 x - 10 y - 32 - 15 x y + 3 y - 35 x + 7 = 0 \Rightarrow 13 x - 7 y - 25 = 0 \Rightarrow 13 x - 7 y = 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3) a l s o \frac{3 x - 15}{x - 9} = \frac{6 y - 5}{2 y + 3} \Rightarrow (3 x - 15) (2 y + 3) = (x - 9) (6 y - 5) \Rightarrow 6 x y + 9 x - 30 y - 45 = 6 x y - 5 x - 54 y + 45 \Rightarrow 6 x y + 9 x - 30 y - 45 - 6 x y + 5 x + 54 y - 45 = 0 \Rightarrow 14 x + 24 y - 90 = 0 \Rightarrow 14 x + 24 y = 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)$
Step 2: solving equation (3) and (4) using elimination method
$E q u a t i o n (3) x 24 \Rightarrow 312 x - 168 y = 600 . . . . . . . . . . . . . . . . . . . . . . . . (5) E q u a t i o n (4) x 7 \Rightarrow 98 x + 168 y = 630 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6) (5) + (6) \Rightarrow 410 x = 1230 \Rightarrow x = \frac{1230}{410} = 3 S u b s t i t u t e x = 3 i n (3) (3) \Rightarrow 13 \times 3 - 7 y = 25 \Rightarrow 39 - 7 y = 25 \Rightarrow 39 - 25 = 7 y \Rightarrow 14 = 7 y \Rightarrow y = 2$
Therefore the values of $x = 3 a n d y = 2$

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