Question:
Solve : (i) (a-b)x (a b)y=a2 -2ab –b2 (a b)(x y)=a2 b2 (ii) ax by – (a-b) = 0 bx – ay – (a b) =0
Answer:
Given,
1. (a - b)x + (a + b)y = a2 - 2ab - b2 .........1
(a + b)(x + y) = a2 - b2
=> (a + b)x + (a + b)y = a2 - b2 .............2
Subtract 1 - 2, we get
=> {(a - b)x + (a + b)y} - {(a + b)x + (a + b)y} = {a2 - 2ab - b2 } - {a2 - b2 }
=> (a - b)x + (a + b)y - (a + b)x - (a + b)y = a2 - 2ab - b2 - a2 - b2
=> (a - b)x - (a + b)x = 2ab - 2b2
=> ax - bx - ax - bx = 2ab - 2b2
=> -2bx = 2b(a - b)
=> -x = a - b
=> x = -(a - b)
=> x = b - a
From equation 1, we get
=> (a - b)*(b - a) + (a + b)y = a2 - 2ab - b2
=> -(a - b)*(a - b) + (a + b)y = a2 - 2ab - b2
=> (a - b)2 + (a + b)y = a2 - 2ab - b2
=> a2 - 2ab + b2 + (a + b)y = a2 - 2ab - b2
=> (a + b)y = a2 - 2ab - b2 - (a2 - 2ab + b2 )
=> (a + b)y = a2 - 2ab - b2 - a2 + 2ab - b2
=> (a + b)y = a2 - b2 - a2 - b2
=> (a + b)y = 2a2 - 2b2
=> (a + b)y = 2(a - b)*(a + b)
=> y = 2(a - b)
So, x = b - a, y = 2(a - b)
2. ax + by - (a - b) = 0
ax + by = (a - b) ...............1
bx – ay - (a + b) = 0
bx – ay = (a + b) ...............2
Multiply equation by a and equation 2 by b and then add, 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
From equation 1, we get
a + by = (a - b)
=> by = a - b - a
=> by = -b
=> y = -b/b
=> y = -1
So, x = 1, y = -1
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