Question

Solve :- ax+by=a-b
bx-ay= a+b
By cross multiplication.

Answer 1

Ax + by = a-b becomes
(ax + by)/(a - b) = 1

bx - ay = a + b becomes
(bx-ay)/(a+b) = 1

So now we can equate the two:

(ax + by)/(a-b) = (bx - ay)/(a+b)
So by cross multiplication:
(ax + by)(a + b) = (bx - ay)(a - b)
a^2x + abx + aby + b^2y = abx - b^2x - a^2y + aby
a^2x + b^2y = -b^2x - a^2y
a^2x + b^2x = -a^2y - b^2y
(a^2 + b^2)x = -(a^2 + b^2)y
x = -y
Substituting into the first equation:
-ay + by = a-b
-(a - b)y = a-b
y = -1
So x = 1

To check: a(1) + b(-1) = a-b, b(1) - a(-1) = a + b

i hope it helps you...
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Answer 2

Given, 

ax + by = a - b   .......1  

bx - ay = a + b   .......2

Multiply by a in equation 1 and b in equation 2, we get

a2 x + aby = a2 - ab   .......3  

b2 x - aby = ab + b2   .......4

Add equation 3 and 4, we get

      a2 x + aby + b2 x - aby = a2 - ab + ab + b2

=> a2 x + b2 x = a2 + b2 

=> x(a2 + b2 ) = a2 + b2

=> x = (a2 + b2 )/(a2 + b2 ) 

=> x = 1

Put value of x in equation 1, we get

=> a + by = a - b

=> by = a - b - a

=> by = -b

=> y = -b/b

=> y = -1

So, x = 1, y = -1