Solving Systems of Equations

 

Three Methods:

1.       Graphical Method

2.       Substitution Method

3.       Addition-Subtraction or Elimination Method

 

Graphical Method

 

In order to solve a system of equations using the graphical method we graph both equations on the same set of axes.  Then we find the point of intersection.

 

Solve the following system of equations using the graphical method. 

       x +  y = 1

       x – 2y = 4

 

Let’s solve each equation for y (to get the equation in slope-intercept form).

 

x + y = 1

Subtract x from each side:                   y = 1 – x

Write equation in slope-intercept form:   y = -x + 1

 

                                                       x – 2y = 4

Subtract x from each side:                   -2y = 4 – x

Divide both sides by –2.                      

Write equation in slope-intercept form:  

 

Graph the equations:

                    

Where do the lines intersect?  At the point (2, -1)  So the solution to our original system of equations is x = 2, y = -1.

 

SUBSTITUTION

 

Steps of solving using the Substitution Method:

1.                Solve one of the equations for one of its variables.

2.                Substitute the expression for the variable found is Step 1 into the other equation.

3.                Find the value of one variable by solving the equation from Step 2.

4.                Find the value of the other variable by substituting the value found in Step 3 into the equation from Step 1.

5.                Check the ordered pair solution in BOTH original equations.

 

Solve the following system of equations using the substitution method: (Same system as before.)

 

       x +  y = 1

       x – 2y = 4

 

1.                (It doesn’t matter which equation we choose.  We just want it to be as easy as possible to solve.  In this problem, it does not matter.)

 

x +  y = 1

x = 1 – y

2.                                 x – 2y = 4

1 – y – 2y = 4

3.                                 1 – 3y = 4

-3y = 3

y = -1

4.                                                                         x = 1 – y

x = 1 – (-1)

                           x = 2

 

5.                         x +  y = 1

2 + (-1) ? 1

1 = 1

This equation checks.

 

x – 2y = 4

                           2 – 2(-1) ? 4

                           2 + 2 ? 4

                           4 = 4

This equation checks.

 

So the solution to this system of equations is x = 2; y = -1

 

Addition-Subtraction Method

 

Steps for solving a system of equations using the Addition-Subtraction Method

1.                Rewrite each equation in standard form, Ax + By = C

2.                If necessary, multiply one or both equations by some nonzero number so that the coefficient of one variable in one equation is the opposite of its coefficient in the other equation.

3.                Add the equations.

4.                Find the value of one variable by solving the equation from Step 3.

5.                Find the value of the second variable by substituting the value found in Step 4 into either original equation.

6.                Check the proposed ordered pair solution in BOTH original equations.

 

Solve the following system of equations using the addition-subtraction method: (Same system as before.)

 

       x +  y = 1

       x – 2y = 4

 

1.                They are already written in standard form.

2.                Let’s eliminate the y’s.  Multiply the first equation by 2.

2(x + y = 1) ฎ  2x + 2y = 2

3.                  2x + 2y = 2

   x – 2y = 4

  3x        = 6

4.                x = 2

5.                x +  y = 1

2 + y = 1

y = -1

6.                x + y = 1

2 + (-1) ? 1

1 = 1

This equation checks.

 

x – 2y = 4

          2 – 2(-1) ? 4

              2 + 2 ? 4

              4 = 4

This equation checks.