IXL - Solve a system of equations in three variables using elimination (Precalculus practice)

Solve the system of equations by elimination.

–
3
x
–
3
y
–
3
z
=
–
18
3
x
+
2
y
+
z
=
–
4
–
2
x
–
y
+
z
=
17

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Learn with an example

question

Solve the system of equations by elimination.

3
x
–
3
y
+
z
=
8
–
2
x
+
2
y
–
z
=
–
4
–
2
x
+
3
y
–
2
z
=
5

key idea

To solve a system of equations with 3 variables by elimination, follow these three steps:

Step 1: Create a system of 2 equations in 2 variables, using elimination.
Step 2: Solve the system of 2 equations in 2 variables.
Step 3: Solve for the remaining variable.

solution

Label the given equations A, B, and C:

A:
3
x
–
3
y
+
z
=
8
B:
–
2
x
+
2
y
–
z
=
–
4
C:
–
2
x
+
3
y
–
2
z
=
5

Step 1: Create a system of 2 equations in 2 variables, using elimination.

Start by creating 2 equations in 2 variables, with the same variable eliminated in each equation.

You can pick any variable to eliminate. To keep the multiplication simple, a good choice here is to eliminate

First, pair equation A with equation B and eliminate

To do that, add the equations.

–

Second, pair equation A with equation C and eliminate the same variable,

To do that, multiply equation A by 2 and add the equations.

(

–

)

–

Step 2: Solve the system of 2 equations in 2 variables.

Now you've reduced the problem to one you know how to solve: 2 equations in 2 variables.

x
–
y
=
4
4
x
–
3
y
=
21

First, pick any variable to eliminate. To keep the multiplication simple, a good choice here is to eliminate

and solve for

To do that, multiply the first equation in 2 variables by

–

and add the equations.

–

(

–

)

–

So,

Second, solve for

To do that, plug

into one of the equations in 2 variables.

–

First equation in 2 variables

–

Plug in

–

Subtract 9 from both sides

–

Subtract 9 from both sides

Multiply both sides by

–

Multiply both sides by

–

So,

Step 3: Solve for the remaining variable.

Finally, solve for the remaining variable,

To do that, plug

and

into one of the given equations in 3 variables.

–

Equation A

(

)

–

(

)

Plug in

and

Plug in

and

–

Multiply

Combine like terms

Subtract 12 from both sides

–

Subtract 12 from both sides

So,

–

The solution to the system of equations is

(

)

(

–

)

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