There are several ways to solve a system of linear equations. This article focuses on 4 methods:
- Graphing
- Substitution
- Elimination: Addition
- Elimination: Subtraction
1. Solve a System of Equations by Graphing
Find the solution to the following system of equations:
y = x + 3
y = -1x - 3
Note: Since the equations are in slope-intercept form, solving by graphing is the best method.
1. Graph both equations.
2. Where do the lines meet? (-3, 0)
3. Verify that your answer is correct. Plug x = -3 and y = 0 into the equations.
y = x + 3
(0) = (-3) + 3
0 = 0
Correct!
y = -1x - 3
0 = -1(-3) - 3
0 = 3 - 3
0 = 0
Correct!
2. Solve a System of Equations by Substitution
Find the intersection of the following equations. (In other words, solve for x and y.)
3x + y = 6
x = 18 -3y
Note: Use the Substitution method because one of the variables, x, is isolated.
1. Since x is isolated in the top equation, replace x in the top equation with 18 - 3y.
3 (18 – 3y) + y = 6
2. Simplify.
54 – 9y + y = 6
54 – 8y = 6
3. Solve.
54 – 8y – 54 = 6 – 54
-8y = -48
-8y/-8 = -48/-8
y = 6
4. Plug in y = 6 and solve for x.
x = 18 -3y
x = 18 -3(6)
x = 18 - 18
x = 0
5. Verify that (0,6) is the solution.
x = 18 -3y
0 = 18 – 3(6)
0 = 18 -18
0 = 0
3. Solve a System of Equations by Elimination (Addition)
Find the solution to the system of equations:
x + y = 180
3x + 2y = 414
Note: This method is useful when 2 variables are on one side of the equation, and the constant is on the other side.
1. Stack the equations to add.
2. Multiply the top equation by -3.
-3(x + y = 180)
3. Why multiply by -3? Add to see.
-3x + -3y = -540
+ 3x + 2y = 414
0 + -1y = -126
Notice that x is eliminated.
4. Solve for y:
y = 126
5. Plug in y = 126 to find x.
x + y = 180
x + 126 = 180
x = 54
6. Verify that (54, 126) is the correct answer.
3x + 2y = 414
3(54) + 2(126) = 414
414 = 414
4. Solve a System of Equations by Elimination (Subtraction)
Find the solution to the system of equations:
y - 12x = 3
y - 5x = -4
Note: This method is useful when 2 variables are on one side of the equation, and the constant is on the other side.
1. Stack the equations to subtract.
y - 12x = 3
0 - 7x = 7
Notice that y is eliminated.
2. Solve for x.
-7x = 7
x = -1
3. Plug in x = -1 to solve for y.
y - 12x = 3
y - 12(-1) = 3
y + 12 = 3
y = -9
4. Verify that (-1, -9) is the correct solution.
(-9) - 5(-1) = -4
-9 + 5 = -4