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Question

Solve for $$x$$ and $$ y$$:
$$(a + 2b) x + (2a - b) y = 2, \\(a - 2b) x + (2a + b)y = 3$$


A
x=1a+115b,y=12a120b
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B
x=12a+15b,y=1a110b
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C
x=1a15b,y=12a+110b
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D
x=12a15b,y=1a+110b
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Solution

The correct option is D $$x=\dfrac{1}{2a}-\dfrac{1}{5b}, y=\dfrac{1}{a}+\dfrac{1}{10b}$$
$$(a+2b)x+(2a-b)y=2$$ ........... (1)
$$(a-2b)x+(2a+b)y=3$$ ............ (2)

Adding (1) and (2), we get

$$2ax+4ay=5$$

$$2x+4y=\dfrac{5}{a}$$ .......... (3)

Subtracting (2) from (1), we get

$$4bx-2by=-1$$

$$4x-2y=-\dfrac{1}{b}$$ ........... (4)

Multiply equation (3) by 2 and subtract (4) from the result,

$$2(2x+4y)-(4x-2y)=\dfrac{10}{a}+\dfrac{1}{b}$$

$$10y=\dfrac{10}{a}+\dfrac{1}{b}$$

$$y=\dfrac{1}{a}+\dfrac{1}{10b}$$ ............ (5)

Now, from (3),

$$2x=\dfrac{5}{a}-4y$$

$$2x=\dfrac{5}{a}-4\left ( \dfrac{1}{a}+\dfrac{1}{10b} \right )$$

$$2x=\dfrac{1}{a}-\dfrac{2}{5b}$$

$$x=\dfrac{1}{2a}-\dfrac{1}{5b}$$ .......... (6)

Mathematics

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