Question

Solve the following pair of linear equation using elimination method:-
(a - b) x + (a + b) y = a² - 2ab - b²
(a + b) (x + y) = a² + b²​

Answer 1

Basic Concept :-

The Elimination Method

• Step 1: Multiply each of the given equation by a suitable constant so that the two equations have the same leading coefficient.

• Step 2: Subtract the second equation from the first so that one variable eliminated.

• Step 3: Solve this new equation to get the value of one variable.

• Step 4: Substitute this value of variable into either of the given equation and solve to get the value of other variable.

Let's solve the problem now!!

Given equations are

$\sf \: (a - b)x + (a + b)y = {a}^{2} - 2ab - {b}^{2} - - (1)$

and

$\sf \: (a + b)(x + y) = {a}^{2} + {b}^{2} - - - (2)$

Now,

Equation (2) can be rewritten as

$\sf \: (a + b)x + (a + b)y={a}^{2}+{b}^{2} - - - (3)$

On Subtracting equation (3) from equation (1), we get

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

$\rm :\longmapsto\: - 2bx = - 2b(a + b)$

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

On substituting x = a + b in equation (1), we get

$\sf \: (a - b)(a + b) + (a + b)y = {a}^{2} - 2ab - {b}^{2}$

$\sf \: {a}^{2} - {b}^{2} + (a + b)y = {a}^{2} - 2ab - {b}^{2}$

$\rm :\implies\:(a + b)y = - 2ab$

$\bf\implies \:y = - \dfrac{2ab}{a + b}$

Additional Information :-

There are 4 methods to solve the pair of linear equations.

• 1. Method of Substitution

• 2. Method of Eliminations

• 3. Method of Cross Multiplication

• 4. Graphical Method

Answer 2

(a-b) × +(a+b) y=a^2-2ab-b^2

a(x+y) +b(y-x) =a^2-2ab-b^2

(a+b) (x+y) =a^2 +b^2

a(x+y) +b(x+y) = a^2+b^2

a(x+y) =a^2+b^2 - b(x+y)

a^2+b^2-b(x+y) +b(y-x) =a^2-2ab-b^2

b^2-bx-by+by-bx= - 2ab-b^2

b^2 =2bx= - 2ab-b^2

b^2+b^2+2ab=2bx..

2b(b+a) =2bx

x=(a+b)

(a+b) (x+y) =a^2+b^2

(a+b) [(a+b)+y]=a^2+b^2

(a+b) ^2+(a+b) y=a^2+b^2

a^2+b^2+2ab+(a+b) y=a^2+b^2

(a+b) y= - 2ab

y=-2ab

a+b