Question

Once you have a system of equations generated by the partial fraction decomposition, can you explain another method to solve it? For example if you hadwe eventually simplify to 7x + 13 = A(3x + 5) + B(x+1). Explain how you could intelligently choose an x-value that will eliminate either A or B and solve for A and B.

Answer 1

Consider the following equation:

7x + 13 = A(3x + 5) + B(x + 1)

 

To explain how intelligently choose an x–value that will eliminate either A or B and solve for A and B. The first method to solve the equation is simply expand and simplify the right side of the equation and then set equal all the corresponding coefficients of both sides. But there is another alternative method to solve the equation is simply eliminate the one term by putting an x-value that would make the coefficient of the term equal to zero.

 

For the equation 7x + 13 = A(3x + 5) + B(x + 1), eliminate A term put the value of x equals to -5/3 so that x = -5/3 and then solve it for B as given below:

7(-5/3) + 13 = A{3(-5/3) + 5} + B{(-5/3) + 1}

   -35/3 + 13 = 0 + B(-5/3 + 1)

               4/3 = -2/3B

                  B = -2

 

Similarly, for the equation 7x + 13 = A(3x + 5) + B(x + 1), to eliminate B term put the value of x equals to -1 i.e. x = -1 and then solve it for A as:

7(-1) + 13 = A{3(-1) + 5} + B{(-1) + 1}

    -7 + 13 = A(-3 + 5) + 0

              6 = 2A

              A = 3

 

Therefore, the solution is:

7x + 13 = 3(3x + 5) – 2(x + 1)

Therefore, the solution is:

7x + 13 = 3(3x + 5) – 2(x + 1)