IXL - Determine the number of solutions to a system of equations in three variables (Algebra 2 practice)

How many solutions does the system of equations below have?

–
x
+
y
–
z
=
–
8
x
–
y
+
z
=
8
–
2
x
+
2
y
–
2
z
=
–
12

no solution

one solution

infinitely many solutions

ref_doc_title.

Learn with an example

question

How many solutions does the system of equations below have?

–
3
x
+
3
y
+
z
=
–
10
–
3
x
+
2
y
–
z
=
8
–
3
x
+
y
–
3
z
=
–
1

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:
–
3
x
+
3
y
+
z
=
–
10
B:
–
3
x
+
2
y
–
z
=
8
C:
–
3
x
+
y
–
3
z
=
–
1

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 A 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.

–
6
x
+
5
y
=
–
2
–
12
x
+
10
y
=
–
31

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.

–

(

–

)

–

The result is

–

a false statement. This means that the two lines — one at the intersection of planes A and B, and one at the intersection of planes A and C — never intersect. So, there is no

solution.

For your reference, here are examples of planes like that. In both diagrams, the lines formed by the intersection of two planes never intersect.

Looking for more? Try this:

$Confetti$

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Work it out

Not feeling ready yet? These can help:

E.13Solve a system of equations in three variables using elimination

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

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