Question

solve for x and you

$(a + b)x + (a - b)y = a^{2} + b {}^{2} \\ \\ (a - b)x + (a + b)y = a {}^{2} + b {}^{2}$
linear equation in 2 variable. 10th ​

Answer 1

Given Equations :

★  (a + b)x + (a - b)y = a² + b² ------------ [1]

★  (a - b)x + (a + b)y = a² + b² ------------ [2]

Multiplying Equation [1] with (a + b), We get :

$:\implies$  (a + b)²x + (a + b)(a - b)y = (a² + b²)(a + b)

$:\implies$  (a + b)²x + (a² - b²)y = (a² + b²)(a + b) -------- [3]

Multiplying Equation [2] with (a - b), We get :

$:\implies$  (a - b)²x + (a + b)(a - b)y = (a² + b²)(a - b)

$:\implies$  (a - b)²x + (a² - b²)y = (a² + b²)(a - b) -------- [4]

Subtracting Equation [4] from Equation [3], we get :

$:\implies$  (a + b)²x + (a² - b²)y - [(a - b)²x + (a² - b²)y] =

                            (a² + b²)(a + b) - (a² + b²)(a - b)

$:\implies$  (a + b)²x + (a² - b²)y - (a - b)²x - (a² - b²)y = (a² + b²)[(a + b) - (a - b)]

$:\implies$  (a + b)²x - (a - b)²x = (a² + b²)[a + b - a + b]

$:\implies$  x[(a + b)² - (a - b)²] = (a² + b²)(2b)

$:\implies$  x[(a² + b² + 2ab) - (a² + b² - 2ab)] = (a² + b²)(2b)

$:\implies$  x[a² + b² + 2ab - a² - b² + 2ab] = (a² + b²)(2b)

$:\implies$  x(4ab) = (a² + b²)(2b)

$\mathsf{:\implies x = \dfrac{(a^2 + b^2)(2b)}{4ab}}$

$\mathsf{:\implies x = \dfrac{a^2 + b^2}{2a}}$

Substituting value of x in Equation [2], We get :

$\mathsf{\implies (a - b) \bigg(\dfrac{a^2 + b^2}{2a}\bigg) + (a + b)y = a^2 + b^2}$

$\mathsf{\implies (a + b)y = a^2 + b^2 - \dfrac{(a^2 + b^2)(a - b)}{2a}}$

$\mathsf{\implies (a + b)y = (a^2 + b^2)\bigg[1 - \dfrac{(a - b)}{2a}\bigg]}$

$\mathsf{\implies (a + b)y = (a^2 + b^2)\bigg[\dfrac{2a - (a - b)}{2a}\bigg]}$

$\mathsf{\implies (a + b)y = (a^2 + b^2)\bigg[\dfrac{2a - a + b}{2a}\bigg]}$

$\mathsf{\implies (a + b)y = (a^2 + b^2)\bigg[\dfrac{a + b}{2a}\bigg]}$

$\mathsf{\implies (a + b)y = \dfrac{(a + b)(a^2 + b^2)}{2a}}$

$\mathsf{\implies y = \dfrac{(a + b)(a^2 + b^2)}{2a(a + b)}}$

$\mathsf{\implies y = \dfrac{a^2 + b^2}{2a}}$

Answers :

$\mathsf{\bigstar\;\;x = \dfrac{a^2 + b^2}{2a}}$

$\mathsf{\bigstar\;\;y = \dfrac{a^2 + b^2}{2a}}$