Question

ax + by = 1 ; bx + ay = (a + b)^2/ a^2 + b^2

Answer 1

ax + by = 1 ------ (1)
bx + ay = [(a + b)^2/(a^2 - b^2)] - 1 ------ (2)

From (2),
bx + ay = [(a + b)(a + b)/(a + b)(a - b)] - 1
bx + ay = [(a + b)/(a - b)] - 1 ------ (3)

Adding (1) and (3),
(x + y)(a + b) = (a + b)/(a - b)
x + y = 1/(a - b)
x = [1/(a - b)] - y ----- (4)

From (4) and (1),
ax + by = 1
[a/(a - b)] - ay + by = 1
y(b - a) = 1 - [a/(a - b)]
y(b - a) = -b/(a - b)
y = b/[(b - a)^2] = [b/(a - b)^2] ----- (5)

From (5) and (4),
x = [1/(a - b)] - y
x = [1/(a - b)] - [b/(a - b)^2]
x = [1/(a - b)] - [b/(a - b)^2]
x = (a - 2b)/(a - b)^2

x = (a - 2b)/(a - b)^2, y = b/(a - b)^2


In case you're wondering, (b - a)^2 = (a - b)^2

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