IXL - Determine the number of solutions to a system of equations in three variables (Precalculus practice)

How many solutions does the system of equations below have?

3
x
+
y
–
2
z
=
–
10
–
x
–
2
y
+
3
z
=
–
2
x
+
y
+
2
z
=
–
12

no solution

one solution

infinitely many solutions

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Learn with an example

question

How many solutions does the system of equations below have?

2
x
+
3
y
+
3
z
=
–
20
–
x
+
y
–
3
z
=
17
–
3
x
–
2
y
–
z
=
–
3

no solution

one solution

infinitely many solutions

key idea

Here are some facts about equations and planes:

A linear equation in three-dimensional space represents a plane. In particular, a linear equation in 3 variables represents a plane.
Two distinct planes are either parallel and don't intersect, or intersect in a line.

To determine the number of solutions for a system of equations with 3 variables, follow the steps below. At the end of each step, try to figure out the number of solutions.

Step 1: Try to create a system of 2 equations in 2 variables.

If you find a false statement (like
0
=
5
),
stop. There is no
solution;
two of the planes do not intersect at all because they are parallel.
If you find a true statement (like
0
=
0
)
— and no false statements! — stop. There are infinitely many
solutions;
at least two of the planes are identical and, they intersect the third in either a line or a plane.
If you find 2 equations in 2 variables (like
2
x
+
3
y
=
6
),
keep
working.
Each pair of planes intersects in a line, but the number of solutions isn't clear yet.

Step 2: Try to solve the system of 2 equations in 2 variables.

If you find a false statement (like
0
=
5
),
stop. There is no
solution;
the lines formed by the intersecting planes in step 1 do not intersect.
If you find a true statement (like
0
=
0
),
stop. There are infinitely many
solutions;
the lines formed by the intersecting planes in step 1 are the same.
If you find 1 equation in 1 variable (like
x
=
3
),
keep
working.
There is one
solution;
the lines formed by the intersecting planes in step 1 intersect in a point, and you can solve for it.

solution

Label the given equations A, B, and C:

A:
2
x
+
3
y
+
3
z
=
–
20
B:
–
x
+
y
–
3
z
=
17
C:
–
3
x
–
2
y
–
z
=
–
3

These can also be called planes A, B, and C.

Step 1: Try to create a system of 2 equations in 2 variables.

Start by trying to create 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.

–

The result is a linear equation in 2 variables. This means that planes A and B intersect to form a line.

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

To do that, multiply equation C by 3 and add the equations.

–

(

–

)

–

The result is a linear equation in 2 variables. This means that planes A and C intersect to form a line.

This step leads to 2 equations in 2 variables. So, continue to step 2.

Step 2: Try to 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
+
4
y
=
–
3
–
7
x
–
3
y
=
–
29

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 7 and add the equations.

(

–

)

–

So,

–

This suggests that there is one solution, a specific point. To be sure, solve for the other variables.

Solve for

To do that, plug

–

into one of the equations in 2 variables.

First equation in 2 variables

–

First equation in 2 variables

(

–

)

Plug in

–

Plug in

–

Multiply

–

Multiply

Add 8 to both sides

So,

Finally, solve for the remaining variable,

To do that, plug

and

–

into one of the given equations in 3 variables.

Equation A

–

Equation A

(

)

(

–

)

Plug in

and

–

Plug in

and

–

Multiply

–

Multiply

Combine like terms

–

Combine like terms

Subtract 4 from both sides

–

Subtract 4 from both sides

Divide both sides by 3

–

Divide both sides by 3

So,

–

Since the planes intersect at a single point — the point

(

)

(

–

)

— there is one

solution.

For your reference, here is an example of planes that intersect at a single point.

Looking for more? Try this:

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Not feeling ready yet? These can help:

N.11Solve a system of equations in three variables using elimination

E.4Find the number of solutions to a system of equations

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