1=A(x-1)+B(x+1) find A and B

Question

1=A(x-1)+B(x+1) find A and B

Answer 1

First, distribute all terms on right side of equation.

 

1 = Ax - A + Bx + B

 

 

Now, we equate coefficients.

 

x:                      A + B = 0

constant:          -A + B = 1

 

 

 

These are the equations you will use to solve for A and B.

 

 A + B = 0                 eq1

-A + B = 1                 eq2

 

 

Add the equations to eliminate the A terms.

 

2B = 1

 

  B = 1/2

 

Then from eq1, we know that

 

A = -1/2 

 

 

To check, we plug them into the original equation.

 

1 = (-1/2)(x - 1) + (1/2)(x + 1)

 

1 = (1/2)[-(x - 1) + (x + 1)]

 

1 = (1/2)(1 - x + x + 1)

 

1 = (1/2)(2)

 

1 = 1

 

 

This checks out!

Answer 2

First, let x = 1 to eliminate A so that you can solve for B:

 

    1 = A(1 - 1) + B(1 + 1)

 

    1 = 2B

 

    B = 1/2

 

Now, let x = -1 to eliminate B so that you can solve for A:

 

    1 = A(-1 - 1) + B(-1 + 1)

 

    1 = -2A

 

    A  = -1/2