Question

X+y=a+b.
ax-by=a²-b²
It's urgent​

Answer 1

$\large\underline{\sf{Solution-}}$

Given pair of linear equations are

$\rm :\longmapsto\:x + y = a + b - - - - (1)$

and

$\rm :\longmapsto\:ax - by = {a}^{2} - {b}^{2} - - - - (2)$

To solve these pair of equations, we use method of Eliminations.

On multiply equation (1) by b, we get

$\rm :\longmapsto\:bx + by = ab + {b}^{2} - - - - (3)$

On adding equation (2) and (3), we get

$\rm :\longmapsto\:bx + ax = ab + {a}^{2}$

$\rm :\longmapsto\:x(a + b) = a(a + b)$

$\bf\implies \:x = a - - - (4)$

On Substituting the value of x in equation (1),

$\rm :\longmapsto\:a + y = a + b$

$\bf\implies \:y = b - - - (5)$

Hence,

Solution Set of equations,

$\rm :\longmapsto\:x + y = a + b$

and

$\rm :\longmapsto\:ax - by = {a}^{2} - {b}^{2}$

is

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underbrace{ \boxed{ \bf \: x = a \: \: \: and \: \: \: y = b}}$

Alternative Method :-

Method of Substitution :

Given equation (1)

$\rm :\longmapsto\:x + y = a + b$

$\rm :\longmapsto\: y = a + b - x - - - (1)$

Given equation (2) is

$\rm :\longmapsto\:ax - by = {a}^{2} - {b}^{2}$

On substituting the value of y, we get

$\rm :\longmapsto\:ax - b(a + b - x) = {a}^{2} - {b}^{2}$

$\rm :\longmapsto\:ax - ba - {b}^{2} + bx = {a}^{2} - {b}^{2}$

$\rm :\longmapsto\:ax - ba + bx = {a}^{2}$

$\rm :\longmapsto\:ax + bx = {a}^{2} + ab$

$\rm :\longmapsto\:x(a + b) = a(a + b)$

$\bf\implies \:x = a$

On substituting the value of x = a in equation (1), we get

$\rm :\longmapsto\: y = a + b - a$

$\bf\implies \:y = b$

Hence,

Solution set is

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underbrace{ \boxed{ \bf \: x = a \: \: \: and \: \: \: y = b}}$