Question

Solve for x and y.

x+y=a+b

ax-by=a^2-b^2

Solve by substitution and elimination method.

Answer 1

Elimination method x + y = a + b ax - by = a² - b² Eliminate y Multiply the first equation thru by b bx + by = ab + b² ax - by = a² - b² Now add them vertically: bx + by = ab + b² ax - by = a² - b² �—-��������—-����������— bx+ax = ab+a² bx + ax = ab + a² Factor out x on the left, factor out a on the right x(b + a) = a(b + a) Divide both sides by (a + b) x(b + a) a(a + b) ���������— = ���������— (a + b) (a + b) Cancel x(b + a) a(a + b) ���������— = ���������— (a + b) (a + b) x = a Now eliminate x Multiply the first equation thru by -a -ax - ay = -a² - ab ax - by = a² - b² Now add them vertically: -ax - ay = -a² - ab ax - by = a² - b² �—-��������—-����������— -ay-by = -ab-b² Multiply thru by -1 ay + by = ab + b² Factor out y on the left, factor out b on the right y(a + b) = b(a + b) Divide both sides by (a + b) y(a + b) b(a + b) ���������— = ���������— (a + b) (a + b) Cancel y(a + b) b(a + b) ���������— = ���������— (a + b) (a + b) y = b Substitution method

(x+y) =(a+b) -------------(1)

ax-by =a²-b² ----------------(2)

Multiplying eq (1) by b and adding (1)and (2)

(bx+by)+(ax-by) = (ab +b2)+(a²-b²)

x(a+b) = ab+a²

x(a+b)= a(a+b)

x=a

Putting this value in eq (1)

a+y=a+b

y=b

Hence x=a and y=b