Solve a system of equations with three variables using elimination

Learn how to solve a system of equations with three variables with one solution using elimination method. We will number the equations in order to make the procedure easy to follow.

Example #1:

1. x - 3y + 3z = -6

2. 2x + 3y - z = 15

3. 4x - 3y - z =  21

The goal is to pick a pair of equations and eliminate a variable by adding the two equations. And then pick another pair of equations and eliminate the exact same variable. Once you have done that, you will be left with a pair of equations with just 2 variables. 

Step 1

Notice that you can easily eliminate y because the y terms are already additive inverse.

Therefore, pair 1. with 2. and 2. with 3. so you can easily get rid of the y terms.

Add the left sides of 1. and 2. and add the right sides of 1. and 2.

You get 3x + 2z = 9 and we call this new equation 4.

1. x - 3y + 3z = -6

2. 2x + 3y - z = 15
________________
4. 3x   +    2z = 9 

Add the left sides of 2. and 3. and add the right sides of 2. and 3.

You get 6x - 2z = 36 and we call this new equation 5.

2. 2x + 3y - z = 15

3. 4x - 3y - z =  21
_________________
5. 6x    -    2z = 36

Step 2

Write the two new equations 4. and 5. as a system. 

4. 3x   +    2z = 9 

5. 6x    -    2z = 36

Solve for x and z. 

Since the z terms are already additive inverses, you can just add the left sides of 4. and 5. and add the right sides of 4. and 5. 

4. 3x   +    2z = 9 

5. 6x    -    2z = 36
_________________
    9x             = 45

x = 5

Use either equation 4. or 5. to find z

4. 3x + 2z = 9

3(5) + 2z = 9

15 + 2z = 9

2z = 9 - 15

2z = -6

z = -3

Step 3

Use either equation 1. 2. or 3. to find y.

2. 2x + 3y - z = 15

2(5) + 3y - -3 = 15

10 + 3y + 3 = 15

13 + 3y = 15

3y = 15 - 13

3y = 2

y = 2/3

The solution of the system is (5, 2/3, -3)


Example #2:

1. 6x - y + 2z = 8

2. 2x + 3y - z = -9

3. 4x + 2y + 5z =  1

Step 1

Multiply equation 2. by 2.

2. 2(2x + 3y - z) = 2(-9)

2. 4x + 6y - 2z = -18

Looking at 1. and 2. you can easily eliminate z because the z terms are now additive inverse.

1. 6x - y + 2z = 8

2. 4x + 6y - 2z = -18

Add the left sides of 1. and 2. and add the right sides of 1. and 2. 

You get 10x + 5y = -10 and we call this new equation 4.

1. 6x - y + 2z = 8

2. 4x + 6y - 2z = -18
__________________
4. 10x + 5y = -10

Multiply equation 2. by 5.

2. 5(2x + 3y - z) = 5(-9)

2. 10x + 15y - 5z = -45

Looking at 3. and 2. you can easily eliminate z because the z terms are now additive inverse.

2. 10x + 15y - 5z = -45

3. 4x + 2y +  5z =  1

Add the left sides of 2. and 3. and add the right sides of 2. and 3. 

You get 14x + 17y = -44 and we call this new equation 5.

2. 10x + 15y - 5z  = -45

3. 4x + 2y + 5z  =  1
____________________
5. 14x + 17y     = -44

Step 2

Write the two new equations 4. and 5. as a system. 

4. 10x + 5y = -10

5. 14x + 17y     = -44

Solve for x and y. 

Multiply 4. by -14 and 5. by 10

4. -14(10x + 5y) = -14(-10)

5. 10(14x + 17y)     = 10(-44)

4. -140x + -70y = 140

5. 140x + 170y     = -440

Since the x terms are now additive inverses, you can just add the left sides of 4. and 5. and add the right sides of 4. and 5. 

4. -140x + -70y = 140

5. 140x + 170y   = -440
______________________
               100y   = -300

y = -3

Use either equation 4. or 5. to find x

4. 10x + 5y = -10

10x + 5(-3) = -10

10x + -15 = -10

10x = 5

x = 5/10

x = 1/2

Use either equation 1. 2. or 3. to find z.

1. 6x - y + 2z = 8

6(1/2) - -3 + 2z = 8

3 + 3 + 2z = 8

6 + 2z = 8

2z = 2

z = 1

The solution of the system is (1/2, -31)

Recent Articles

  1. Box and Whiskers Plot

    Nov 18, 22 08:20 AM

    Easily learn to construct a box and whiskers plot for a set of data by using the median and the extreme values.

    Read More

  2. Binary Number System

    Nov 17, 22 10:53 AM

    This lesson will give you a deep and solid introduction to the binary number system.

    Read More

Tough algebra word problems

100 Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

Math quizzes

 Recommended

Math vocabulary quizzes