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3 Variable Systems (copy) - Printable Mathematics Worksheets - Wayground

3 Variable Systems (copy)

20 questions

9th - 12th Grade

Mathematics

This free printable mathematics worksheet is designed for Grades 9–12 and contains 20 multiple-choice questions on systems of linear equations in two and three variables. Students identify coefficients, distinguish ordered pairs from ordered triples, interpret solutions as intersections of lines or planes, determine whether a system has one, none, or infinitely many solutions, and verify candidate solutions. The worksheet also asks students to translate age, animal, and number-word problems into systems of equations before solving them. These varied items build accuracy with algebraic representation, solution classification, and coordinate order while checking both conceptual understanding and procedural reasoning. A complete answer key is included, making the worksheet useful for independent practice, review, homework, or formative assessment.

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Worksheets

3 Variable Systems (copy)

Total questions: 20

Name
Class
Date
1.

Which of the following is NOT a coefficient of the following equation: 4z+5x-y=0

a)

5

b)

4

c)

-1

d)

0

2.
The solution of a system of three linear equations in three variables, x,y, and z, is called a(n) ___________________ (x,y,z).
a)
Ordered Pair
b)
Integer
c)
Ordered Triple
d)
Sum
3.
The solution of a system of linear equations is...
a)
the y-intercept (b)
b)
the intersection of the two equations on a graph
c)
the second equations' answers
d)
the slope (m)
4.
What is the solution?
a)
(1, -1)
b)
(-1, 1)
c)
(0, -2)
d)
(2, 0)
5.
This system has _____ solutions
a)
0
b)
1
c)
2
d)
Infinitely many
6.

The solution to a 3-variable systems of equation is...

a)

the intersection of three planes

b)

the origin

c)

infinitely many solutions

d)

no unique solution

7.
Erin is 3 years younger than twice Alex's age. Their ages combined are 33 years. How old are Alex and Erin. If x=Erin's age and y=Alex's age, choose the system that matches the situation.
a)
x + y = 33
y = 2x - 3
b)
x + y = 33
x = 2y - 3
c)
x + y = 33
x = 3 - 2y
d)
x + y = 3
x = 33 - 2y
8.
There are 15 animals in a barn.  These animals are horses and chickens.  There are 44 legs in all.  Which system of equations represents the situation?
a)
x + y = 15
4x + 2y = 44
b)
4x + 2y = 15
x + y = 44
c)
x = 2y + 44
4x = y + 15
d)
2x - 4y = 44
x - y = 15
9.
The sum of two numbers is 14. The larger number is 2 less than three times the smaller number. Which system could be solved to find the value of these numbers?
a)
x+y=14
y=3x-2
b)
x+y=2
3x-2y=14
c)
x+y=14
y=2-3x
d)
x+y=14
y=3x+2
10.
Solve the system of equations: 
a)
( 4, 2, -1)
b)
( 4, 3, -1)
c)
None of these
d)
( -4, 3, 1)
11.

Solve the following system of equations.

a)

(15, 120, 45)

b)

(120, 15, 45)

c)

(15, -15, 0)

d)

(-15, 15, 0)

12.

Is (4,5,6) a solution to 2x-3y+z=8

a)

Yes

b)

No

13.

What is your solution if your variables go away and you are left with 0 = 0?

a)

No solution

b)

Infinitely many solutions

c)

All real numbers

d)

(x,y,z)

14.

What is your solution if your variables go away and you are left with 0 = 17?

a)

No unique solution

b)

Infinitely many solutions

c)

No solution

d)

(x,y,z)

15.

How many solutions does the following system of equations have?

a)

No Solution

b)

Infinitely Many Solutions

c)

One Solution

d)

Cannot Be Determined

16.

How many solutions does the following system of equations have?

a)

No Solution

b)

Infinitely Many Solutions

c)

One Solution

d)

Cannot Be Determined

17.

How many solutions does the following system of equations have?

a)

No Solution

b)

Infinitely Many Solutions

c)

One Solution

d)

Cannot Be Determined

18.

A student in solving the following system of equations using the elimination method. Explain what their first step would be if they wanted to eliminate the y variable from equations (i) and (ii).

a)

They would multiply equation (i) by 3 and equation (ii) by 4.

b)

They would multiply equation (iiI) by 4.

c)

They would multiply equation (i) by -3 and multiply equation (ii) by 4

d)

They would multiply equation (i) by 3

19.

A student was solving the system by substitution and has shown the work below. In which part did the student make a mistake?

a)

Part 1

b)

Part 2

c)

Part 3

20.

How confident do you feel solving linear systems in three variables?

a)

Very confident, and could help a friend.

b)

Fairly confident, but would like more practice

c)

Somewhat confident, but have some questions

d)

Not confident yet, and would like some help

Answer Key

3 Variable Systems (copy)

Total questions: 20

1.
d)

0

2.
c) Ordered Triple
3.
b) the intersection of the two equations on a graph
4.
a) (1, -1)
5.
a) 0
6.
a)

the intersection of three planes

7.
b) x + y = 33
x = 2y - 3
8.
a) x + y = 15
4x + 2y = 44
9.
a) x+y=14
y=3x-2
10.
b) ( 4, 3, -1)
11.
a)

(15, 120, 45)

12.
b)

No

13.
b)

Infinitely many solutions

14.
c)

No solution

15.
a)

No Solution

16.
b)

Infinitely Many Solutions

17.
c)

One Solution

18.
a)

They would multiply equation (i) by 3 and equation (ii) by 4.

19.
c)

Part 3

20.
n/a

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FAQs

What does this 3-variable systems worksheet cover?

This 20-question multiple-choice worksheet practices interpreting and solving systems of linear equations in two and three variables. Students identify coefficients, use ordered triples, interpret intersections of lines and planes, classify systems as having one, no, or infinitely many solutions, translate word problems into equations, and check whether a coordinate set satisfies a system.

How can I use this worksheet to teach systems of linear equations in three variables?

Introduce the vocabulary first, including coefficients, ordered triples, and the idea that a three-variable solution represents the intersection of three planes. Then have students model the age, animals, and number problems by defining variables and writing equations before attempting the solution items. Use the questions about 0 = 0 and 0 = 17 to prompt discussion about why a system has infinitely many solutions or no solution.

What mistakes should I watch for when students complete this systems worksheet?

Watch for students selecting 0 as a coefficient when the equation contains a variable term with a coefficient of -1, or confusing an ordered pair with an ordered triple. In the word problems, students may reverse the variables or misread relationships such as “twice an age minus 3,” and in verification items they may fail to substitute every coordinate. Students may also confuse 0 = 0, which indicates infinitely many solutions, with 0 = 17, which indicates no solution.

How do I assign and grade this systems of equations worksheet?

This worksheet is available as a printable PDF and as a digital quiz on Wayground, with a complete answer key for all 20 multiple-choice questions. Grade printable submissions by scanning student work with the Wayground for Teachers app, or use the digital quiz for online response collection and grading. The printable format can also support schools that are reducing screen time.

How can I differentiate this systems of equations worksheet for my students?

Use the worksheet’s Advanced Settings to adjust font spacing and font size so the multiple-choice systems, coordinate triples, and word-problem equations are easier to read. You can also apply a dyslexia-friendly font or translate the worksheet into another language when that supports access to the algebra vocabulary and problem statements.

Where can I find more free worksheets like this on Wayground?

Wayground offers a comprehensive collection of free printable worksheets and practice problems across subjects, with downloadable PDFs and answer keys to help students master essential skills and concepts. Browse more resources at https://wayground.com/en-us/worksheets.

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