3.2 Solving Linear Systems Algebraically

 

There are two different ways to solve a linear system:  The substitution method and the linear combination method.

What is a linear system? A linear system is a system of linear equations:

Ex)  2x+4y=12

          x+2y=6

 

Using the combination method to solve linear systems

Equation 1 3x+5y=16

Equation 2 3x-2y=6

 

1) In this case you must multiple 1 equation by -1 so that one of the coefficients in 1 of the equations is opposite of the other coefficient in the other equation.

 

Equation 1=  (-1)(3x+5y=-16)

                       -3x-5y=16

 

2) Then add the new equation to equation 2.  Hence the name the combination method.

 

                    -3x-5y=16

                  + 3x-2y=-9

                         -7y = 7

                         -7     -7

                           Y=-1

 

3) Once you have found your y value plug it into either of the original equations to find the x value.

 

              3x-2(-1)=-9

                   3x+2=-9

                       3x=-11

                        X=-11/3

 

Your values are (-11/3,-1)

 

4) The final step is to plug in the values into both of the original equations.  If the solutions are true our answers are correct.

 

The two solutions x and y are where the two linear equations would intercept on a a graph.  If for some reason your math doesn’t work out or you can not find a true solution to your linear system it does not necessarily mean you are wrong it means that the two lines to not intersect.