# How do you solve the system of linear equations: (a+2b)x+(2a-b)y=2 and (a-2b)x+(2a+b)y=3?

Sep 30, 2017

$x = \frac{1}{2 a} - \frac{1}{5 b}$ and $y = \frac{1}{a} + \frac{1}{10 b}$

#### Explanation:

We have $\left(a + 2 b\right) x + \left(2 a - b\right) y = 2$ ...................(1)

$\left(a - 2 b\right) x + \left(2 a + b\right) y = 3$ ...................(2)

Adding the two we get $2 a x + 4 a y = 5$ ...................(3)

and subtracting (2) from (1), we get

$4 b x - 2 b y = - 1$ ...................(4)

Multiplying (3) by $2 b$ and (4) by $a$, we get

$4 a b x + 8 a b y = 10 b$ ...................(5)

and $4 a b x - 2 a b y = - a$ ...................(6)

Subtracting (6) from (5), we have

$10 a b y = 10 b + a$ or $y = \frac{10 b + a}{10 a b} = \frac{1}{a} + \frac{1}{10 b}$

Multipying (6) by $4$ and adding to (5), we get

$20 a b x = 10 b - 4 a$ or $x = \frac{10 b - 4 a}{20 a b} = \frac{1}{2 a} - \frac{1}{5 b}$