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

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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