Solving a System of Equations

We use a system of equations to solve for more than one variable.

EXAMPLE –>  { 2x = y - 4 }

                        { 3x + y = 9}

There are multiple ways to solve the system.

One way is by using substitution.

STEP 1: one of the equations has to be set equal to x or y

            We will use the top equation 2x = y - 4 and set the it equal to y               2x = y - 4

            To get y alone we have to add 4 to both sides                                 (2x) + 4 = (y - 4) + 4

            We are left with the system as                                                           {2x + 4 = y}

                                                                                                                       {3x + y = 9}

STEP 2: We now have to substitute the equation we set equal to y into 

            the second equation to make one equation with one variable

                                                                                                                        3x + (2x + 4) = 9

                                                                                                                                  5x + 4 = 9

            Now we have to solve for x                                                                    (5x + 4) - 4 = (9) - 4

                                                                                                                                         5x = 5

                                                                                                                                          5     5   

                                                                                                                                           x = 1

STEP 3: We then take x and plug it back into one of the equations to find y

                                                                                                                            y = 2(1) + 4

                                                                                                                            y = 8

The second way is elimination

STEP 1: We have to make the two equations look similar                                    {2x = y - 4}  ->   {2x - y = -4}

                                                                                                                              {3x + y = 9}        {3x +y = 9 }

            Now we can add the two systems together                                                                     5x + (0y) = 5

                                                                                                                                                               5x = 5

                                                                                                                                                                5     5

                                                                                                                                                                 x = 1

STEP 2: plug x back into one of the equations.

                                                                                                                                                    2 • (1) = y - 4

                                                                                                                                                   (2) + 4 = (y - 4) +4

                                                                                                                                                            6 = y      

The third way to solve a system is by using a graphing utility.

STEP 1: graph one of the equations –>  2x = y - 4

STEP 2: graph the second equation –>  3x + y = 9

STEP 3: Find where the two lines intersect on the same plane

The lines cross at (1,6). x = 1 and y = 6