Question

Solve by elimination method ax +by-a+b=0. bx-ay-4a-b=0

Answer 1

Hey Mate !

Here is your solution :

Well, we need to know , what is elimination method ?

It is a method that we use to find the value of variables , if any linear equation is in two variables and 2 equations are given.

Steps :

1. Eliminate any one of the two variables.

2. Substitute the value of the variable that you got in any of the equations.

Moving to your question.

ax + by - a + b = 0 -------- ( 1 )

bx - ay - 4a - b = 0 ----------- ( 2 )

According to rule,

We have to first eliminate any of the variables.

Let's go for y.

Coefficient of y in ( 1 ) = b

Coefficient of y in ( 2 ) = -a

By multiplying in ( 1 ) by a and in ( 2 ) by b.

=> a ( ax + by - a + b ) = 0 ×a

=> a²x + aby - a² + ab = 0 -------- ( 3 )

And,

=> b ( bx - ay - 4a - b ) = 0 × b

=> b²x - aby - 4ab - b² = 0 -------- ( 4 )

Adding ( 3 ) and ( 4 ),

=> a²x + aby - a² + ab = 0

=> b²x - aby - 4ab - b² = 0
__________________________

=> a²x + b²x - a² - 4ab - b² + ab = 0

Rearranging the terms :

=> a²x + b²x - a² - b² - 4ab + ab = 0

Taking x as common ,

=> x ( a² + b² ) - ( a² + b² ) - 4ab + ab = 0

=> x ( a² + b² ) - ( a² + b² ) - 3ab = 0

Taking out ( a² + b² ) as common,

=> ( a² + b² ) ( x - 1 ) - 3ab = 0

=> ( a² + b² ) ( x - 1 ) = 3ab

=> ( x - 1 ) = 3ab / ( a² + b² )

=> x = { 3ab / ( a² + b² ) } + 1

=> x = ( 3ab + a² + b² ) / ( a² + b² )

By substituting the value of x in ( 1 ),

=> ax + by - a + b = 0

=> a { ( a² + b² + 3ab ) / ( a² + b² ) } + by - a + b = 0


=> { ( a³ + ab² + 3a²b ) / ( a² + b² ) } + by - a + b = 0



=> { a³ + ab² + 3a²b + by( a² + b² ) - a( a² + b² ) + b ( a² + b² ) } / ( a² + b² ) = 0

=> a³ + ab² + 3a²b - a³ - ab² + a²b + b³ + by ( a² + b² ) = 0 × ( a² + b² )

=> 4a²b + b³ + by ( a² + b² ) = 0

=> by ( a² + b² ) = - 4a²b - b³

=> by ( a² + b² ) = - ( 4a²b + b³ )

=> by = - ( 4a²b + b³ ) / ( a² + b² )

=> y = - ( 4a²b + b³ ) / b ( a² + b² )

Taking out b as common,

=> y = - b( 4a² + b² ) / b ( a² + b² )

=> y = - ( 4a² + b² ) / ( a² + b² )

So,

x = ( a² + b² + 3a²b ) / ( a² + b² )

y = - ( 4a² + b² ) / ( a² + b² )

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Hope it helps !!