Solve a system of equations with three variables using substitution

Learn how to solve a system of equations with three variables with one solution using substitution 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

Step 1

Choose 1. and solve for x.

x - 3y + 3z = -6

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

x + 3z = -6 + 3y

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

x = -6 + 3y - 3z

Step 2

Substitute -6 + 3y - 3z for x into 2. and 3.

Substitute -6 + 3y - 3z for x into 2.

2x + 3y - z = 15

2(-6 + 3y - 3z) + 3y - z = 15

-12 + 6y - 6z + 3y - z = 15

-12 + 9y - 7z = 15

4. 9y - 7z = 27

Substitute -6 + 3y - 3z for x into 3.

4x - 3y - z =  21

4(-6 + 3y - 3z) - 3y - z =  21

-24 + 12y - 12z - 3y - z = 21

-24 + 9y - 13z = 21

5. 9y - 13z = 45

Step 3

Write the two new equations as a system and solve for y and z

4. 9y - 7z = 27

5. 9y - 13z = 45

Multiply 5. by -1

-1(9y - 13z) = -1(45)

-9y + 13z = -45

4. 9y - 7z = 27

5. -9y + 13z = -45

Add the left sides of 4. and 5. and add the right sides of 4. and 5.

4. 9y - 7z = 27

5. -9y + 13z = -45
_________________
            6z    = -18

z = -3

Substitute the value of z in 4. to get y 

4. 9y - 7z = 27

9y - 7(-3) = 27

9y + 21 = 27

9y = 6

y = 6/9

y = 2/3

Step 4

Substitute the values for y and z into one of the original equations and solve for x. Let us replace in 1.

1. x - 3y + 3z = -6

x - 3(2/3) + 3(-3) = -6

x - 2 + -9 = -6

x + -11 = -6

x = -6 + 11

x = 5

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

Example #2:

1. 2x + y - z = 5

2. x + 4y + 2z = 16

3. 15x + 6y - 2z =  12

Step 1

Choose 1. and solve for z.

2x + y - z = 5

2x + y - z + z = 5 + z

2x + y - 5 = 5 - 5 + z

2x + y - 5 = z

Step 2

Substitute 2x + y - 5 for z into 2. and 3.

Substitute 2x + y - 5 for z into 2.

x + 4y + 2z = 16

x + 4y + 2(2x + y - 5) = 16

x + 4y + 4x + 2y - 10 = 16

5x + 6y = 16 + 10

4. 5x + 6y = 26

Substitute 2x + y - 5 for z into 3.

15x + 6y - 2z =  12

15x + 6y - 2(2x + y - 5) =  12

15x + 6y - 4x - 2y + 10 = 12

11x + 4y = 12 - 10

5. 11x + 4y = 2

Step 3

Write the two new equations as a system and solve for x and y

4. 5x + 6y = 26

5. 11x + 4y = 2

Multiply 4. by -2 and 5. by 3

4. -2(5x + 6y) = -2(26)

5. 3(11x + 4y) = 3(2)

4. -10x + -12y = -52

5. 33x + 12y = 6

Add the left sides of 4. and 5. and add the right sides of 4. and 5.

4. -10x + -12y = -52

5. 33x + 12y   = 6
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   23x              = -46

x = -2

Substitute the value of x in 4. to get y 

4. 5x + 6y = 26

5(-2) + 6y = 26

-10 + 6y = 26

6y = 36

y = 6

Step 4

Substitute the values for x and y into one of the original equations and solve for z. Let us replace in 1.

1. 2x + y - z = 5

2(-2) + 6 - z = 5

-4 + 6 - z = 5

2 - z = 5

2 - 5 = z

z = -3

The solution of the system is (-2, 6, -3)