Solve: x+y=a+b and ax-by = a^2-b^2

Solve: x+y=a+b and ax-by = a^2-b^2 by Elemination Method and Substitution Method.

Elemination Method

Solve: x+y=a+b and ax-by = a^2-b^2

x+y=a+b –(i)

ax-by = a²-b² –(ii)

Multiplying qequation (i) by b we get,

bx+by = ab+b² — (iii)

Now adding equation (ii) and (iii) we get,

ax-by+bx+by =a²-b² +ab+b²

⟹ x(a+b) = a(a+b)

⟹ x = a

Substituting the value of x in equation (i) we get,

a+y= a+b

⟹ y=b

Therefore Required Solutions are x = a and y =b.

Substitution Method

x+y=a+b –(i)

ax-by = a²-b² –(ii)

From Equation (i) we get,

x=a+b-y —-(iii)

Now Substituting the value of x in equation (ii) we get,

a(a+b-y)-by=a²-b²

⟹ a²+ab-ay-by=a²-b²

⟹ ab-y(a+b)=a²-b² -a²

⟹ ab-y(a+b)= -b²

⟹ ab+b² =y(a+b)

⟹ b(a+b)=y(a+b)

⟹ y = b

From Equation (iii) we get,

x =a+b-b

⟹ x = a

Required Solutions are x = a and y = b .