Question

solve for x and y: ax + by = b - a and bx - ay= - (a + b)

Answer 1

Answer:

$\boxed{\sf \: \: x = - 1 \: \: and \: \: y = 1 \: }$

Step-by-step explanation:

Given pair of linear equation is

$\sf \: ax + by = b - a - - - (1)$

and

$\sf \: bx - ay = - (a + b) - - - (2)$

On multiply equation (1) by a and (2) by b, we get

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

and

$\sf \: {b}^{2} x - aby = - ab - {b}^{2} - - - (4)$

On adding equation (3) and (4), we get

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

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

$\implies\sf \: x = - 1$

On substituting x = - 1 in equation (1), we get

$\sf \: a( - 1) + by = b - a$

$\sf \: - a + by = b - a$

$\sf \: by = b$

$\implies\sf \: y = 1$

Hence,

$\implies\sf \: x = - 1 \: \: and \: \: y = 1$

$\rule{190pt}{2pt}$

Concept Used :-

There are 4 methods to solve this type of pair of linear equations.

1. Method of Substitution

2. Method of Eliminations

3. Method of Cross Multiplication

4. Graphical Method

I prefer here Method of Eliminations :-

To solve systems using elimination, follow this procedure:

The Elimination Method

Step 1: Multiply each equation by a suitable number so that the two equations have the same leading coefficient.

Step 2: Subtract the second equation from the first to eliminate one variable and get new equation in one variable.

Step 3: Solve the new equation to get a value of variable.

Step 4: Substitute the value of variable thus evaluated into either Equation 1 or Equation 2 and get the value of other variable.