Question

11. Solve by cross multiplication: x+y = a+b and ax-by= a^2 - b^2.

Answer 1

→ x + y - (a + b) = 0

→ ax - by - (a² - b²) = 0

$\mathsf{= \frac{x}{ - ( {a}^{2} - {b}^{2}) - b(a + b)}} \\ \mathsf{= \frac{x}{ - {a}^{2} + {b}^{2} - ab - {b}^{2} }} \\ \mathsf{= \frac{x}{ - a(a + b)}}$

$\mathsf{ = \frac{ - y}{ - ( {a}^{2} - {b}^{2} ) + a(a + b) }} \\ \mathsf{= \frac{ - y}{ - {a}^{2} + {b}^{2} + {a}^{2} + ab }} \\ \mathsf{= \frac{ - y}{b(a + b)}}$

$\mathsf{ = \frac{1}{ - b - a}} \\ \mathsf{= \frac{1}{- (a + b)}}$

______________________________

$\mathsf{\frac{x}{-a(a+b)} = \frac{-y}{b(a+b)} = \frac{1}{-(a + b)}} \\ \mathsf{x = \frac{-a(a+b)}{-(a+b)} = a} \\ \mathsf{-y = \frac{b(a+b)}{-(a+b)} = b}$

x = a & y = b