6(ax+by) =3a+2b , 6(bx-ay)=3a-2b

6(ax + by) = 3a + 2b
6ax + 6by = 3a + 2b.........(i)
6(bx - ay) = 3b - 2a
6bx - 6ay = 3b - 2a.........(ii)

Multiplying (i) by a and (2) by b,
So, 6a²x + 6aby = 3a² + 2ab .......... (iii)
And 6b²x - 6aby = 3b² - 2ab ......... (iv)
Add (iii) and (iv), we get
6a²x + 6aby + 6b²x - 6aby = 3a² + 2ab + 3b² - 2ab
⇒ 6a²x + 6b²x = 3a² + 3b²
⇒6 (a²x + b²x) = 3(a² + b²)
⇒{tex}x = \frac { 3 \left( a ^ { 2 } + b ^ { 2 } \right) } { 6 \left( a ^ { 2 } + b ^ { 2 } \right) } = \frac { 3 } { 6 } = \frac { 1 } { 2 }{/tex}

Substituting {tex}x = \frac 12{/tex} in (i),we get
{tex}6 a \times \frac { 1 } { 2 } + 6 b y = 3 a + 2 b{/tex}
⇒ 3a + 6by = 3a + 2b
⇒ 6by =3a + 2b -3a
⇒ 6by = 2b
⇒ {tex}y = \frac { 2 b } { 6 b } = \frac { 1 } { 3 }{/tex}
Hence, the solution is {tex}x = \frac { 1 } { 2 } , y = \frac { 1 } { 3 }{/tex}