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How to Solve a System of Linear Equations

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There are several ways to solve a system of linear equations. This article focuses on 4 methods:

  1. Graphing
  2. Substitution
  3. Elimination: Addition
  4. 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!

Systems of Linear Equations Worksheet

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

Systems of Linear Equations Worksheet

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

Systems of Linear Equations Worksheet

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

Systems of Linear Equations Worksheet

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