Question

Solve the following system of linear equation
2(ax - y) + + 4) = 0,2(bx + ay) + (b - 4a) = 0​

Answer 1

Appropriate Question :- .

Solve the following system of linear equations :-

• 2(ax - by) + a + 4b = 0

• 2(bx + ay) + b - 4a = 0

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

Given pair of linear equations are

$\rm :\longmapsto\:2(ax - by) + a + 4b = 0 - - - (1)$

and

$\rm :\longmapsto\:2(bx + ay) + b - 4a = 0 - - - (2)$

Now,

☆ Equation (1) can be rewritten as

$\rm :\longmapsto\:2ax - 2by = - a - 4b - - - ( 3 )$

and

☆ Equation (4) can be rewritten as

$\rm :\longmapsto\:2bx + 2ay = 4a - b - - - (4)$

Now,

☆ Multiply equation (3) by b and equation (4) by a, we get

$\rm :\longmapsto\:2abx - 2 {b}^{2} y = - ab - 4 {b}^{2} - - - (5)$

and

$\rm :\longmapsto\:2abx + 2 {a}^{2} y = 4 {a}^{2} - ab - - - (6)$

Now,

☆ On Subtracting equation (6) from equation (5), we get

$\rm :\longmapsto\: - {2b}^{2}y - {2a}^{2} y = - {4a}^{2} - {4b}^{2}$

$\rm :\longmapsto\: - 2y({b}^{2} + {a}^{2}) = - 4( {a}^{2} + {b}^{2} )$

$\purple{\bf\implies \:y = 2}$

☆ On substituting y = 2, in equation (3) we get

$\rm :\longmapsto\:2ax - 4b = - a - 4b$

$\rm :\longmapsto\:2ax = - a$

$\rm :\longmapsto\:2x = - 1$

$\purple{\bf\implies \:x = \: - \: \dfrac{1}{2}}$

Basic Concept :-

☆ 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

We prefer here Method of Elimination :-

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 of the two variables.

• Step 3: Solve this new equation for one variable.

• Step 4: Substitute the value of this variable into either of the given two equations to get the value of other variable.