Solve the system of equations.

x – 2y = 2
3x + y = 6

Graphing Method
To graphically solve a system of equations, graph both equations on the same coordinate plane and find the point of intersection. This point (x, y) is the solution because it is the point that lies on both lines and will make both equations true.

To begin, rewrite both equations in slope-intercept form (y = mx + b).

Equation 1: x – 2y = 2
 
  X minus 2 times y equals 2. Subtract x from both sides.
–2y
=
–x + 2

–2y over –2
=
–x over –2 + 2 over –2

y
=
1 over 2x – 1


slope = 1 over 2 and y-intercept = –1		 Equation 2: 3x + y = 6
 
   3 times x plus y equals 6. Subtract the term 3 times x from both sides of the equation.
y
=
–3x + 6

slope = –3 over 1 and y-intercept = 6

Now that you know the slope and y-intercept of each equation, graph both lines on a coordinate plane and find the point of intersection. Recall that you will plot the y-intercept first, and then use the slope to find a second point.

coordinate graph with 2 equations graphed. First line passes through the points (0, -1) and (2, 0). The second line passes through the point s (0, 6) and (1, 3).
You can see that the point of intersection is the ordered pair (2, 0). The final step of this process is to check your solution in both equations. Substitute 2 for x and 0 for y.



Equation 1:

x – 2y
=
2

2 – 2(0)
question mark above an equals sign
2

2 – 0
question mark above an equals sign
2

2
=
2 check mark
Equation 2:

3x + y
=
6

3(2) + 0
question mark above an equals sign
6

6 + 0
question mark above an equals sign
6

6
=
6 check mark
Substitution Method
Now solve the same system of equations using the substitution method.
  • Isolate one of the variables in one of the equations.

    Using the second equation, isolate the y.

    3 times x plus y equals 6. Subtract the term 3 times x from both sides of the equation.
     
    y = –3x + 6

  • Substitute the expression for the isolated variable into the other equation and solve for the other variable.

    In this case –3x + 6 will be substituted for y in the first equation.

    Select to reveal Answer.

  • Substitute the value of the first variable into one of the original equations and solve for the second variable.

    Select to reveal Answer.

  • Check your solution in both equations. The solution must make both equations true.

    Select to reveal Answer.

Finally, review solving systems of equations using the elimination method. Remember, this is just another way of getting to the same solution.

Elimination Method
  • Look at the variables in pairs to determine if a variable will cancel out by combining the equations.

    x – 2y
    =
    2

    3x + y
    =
    6


    Select to reveal Answer.

  • If a variable will automatically cancel out, combine the two equations. If a variable will not cancel out, choose a variable to eliminate.

    One pair of variables must have the same coefficient with the opposite sign. Since the coefficients of y already have opposite signs, they will be easier to eliminate.

  • Multiply one or both of the equations by a number(s) to create a set of equations where a variable cancels out.

    Select to reveal Answer.

  • Combine the equations and solve for the remaining variable.

    Select to reveal Answer.

  • Substitute the value for the first variable into either of the original equations and solve for the second variable.

    Select to reveal Answer.

  • Check your solution in both equations. If you have the correct solution, it will make both equations true.

    Select to reveal Answer.

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