Sample Worksheet

Solving Systems of Linear Equations

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Material Overview

Graphing, substitution, and elimination

Subject Mathematics
Subject Area Algebra I

Grade/Level 8th Grade

Type Worksheet

Difficulty Medium

Learning Setting Tutoring

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Teacher Context

Teacher Lesson Scope

This worksheet provides guided practice for solving systems of linear equations using graphing, substitution, and elimination methods. Students must have prerequisite knowledge of graphing linear equations in slope-intercept form and solving single-variable multi-step equations. The practice progresses from highly structured, scaffolded exercises to independent multi-step problems and real-world applications.

Printable Student Copy

Student Version

Name:
Date:

Directions: Complete each section of this worksheet independently. Read the explanations and the worked example, then use the provided workspaces and fill-in-the-blank steps to solve each problem.

Worked Example for Reference:
Solve the system of equations:
y = 2x - 1
y = -x + 5

Since both equations are equal to y, set them equal to each other to solve for x:
2x - 1 = -x + 5
Add x to both sides:
3x - 1 = 5
Add 1 to both sides:
3x = 6
Divide by 3:
x = 2

Now, substitute x = 2 back into the first equation to find y:
y = 2 × 2 - 1
y = 4 - 1
y = 3

The solution is the ordered pair (2, 3).
To check the solution, substitute x = 2 and y = 3 into both original equations:
Equation 1: 3 = 2 × 2 - 1 (This simplifies to 3 = 3, which is true)
Equation 2: 3 = -2 + 5 (This simplifies to 3 = 3, which is true)

  1. Which ordered pair is the solution to the system of equations graphed below?
    Equation 1 is a line that passes through the points (0, 1) and (1, 3).
    Equation 2 is a line that passes through the points (0, 5) and (1, 3).
    A. (0, 1)
    B. (1, 3)
    C. (0, 5)
    D. (2, 5)

  2. Consider the system of equations below:
    y = -2x + 4
    y = -2x - 1
    How many solutions does this system of equations have?
    A. Exactly one solution
    B. No solution
    C. Infinitely many solutions
    D. Exactly two solutions

  3. A rental company offers two plans. Plan A costs a flat fee of 20 dollars plus 2 dollars per hour. Plan B costs a flat fee of 10 dollars plus 4 dollars per hour. Which system of equations represents the cost, y, for x hours of use?
    A. y = 20x + 2 and y = 10x + 4
    B. y = 2x + 20 and y = 4x + 10
    C. y = 22x and y = 14x
    D. y = 2x - 20 and y = 4x - 10

  4. What is the most logical first step to solve the system of equations below using substitution?
    x = y + 3
    2x + 3y = 11
    A. Substitute y + 3 for x in the second equation.
    B. Substitute 2x + 3y for x in the first equation.
    C. Add the two equations together to eliminate x.
    D. Subtract 3 from both sides of the second equation.

  5. Solve the system of equations using substitution by filling in the missing steps:
    y = 3x
    2x + y = 15

Step 1: Substitute 3x for y in the second equation:
2x + = 15

Step 2: Combine like terms:
5x = 15

Step 3: Solve for x:
x =

Step 4: Substitute the value of x back into the first equation to find y:
y = 3 ×
y = 9

  1. Solve the system of equations using elimination by filling in the missing steps:
    4x + y = 13
    2x - y = 5

Step 1: Add the two equations together to eliminate y:
6x =

Step 2: Solve for x:
x =

Step 3: Substitute the value of x back into the first equation to solve for y:
4 × + y = 13
12 + y = 13
y =

  1. Solve the system of equations using any method of your choice. Show your work and check your final answer in both equations.
    x + y = 8
    y = 3x

Final Solution:
x =
y =

  1. Two different fitness clubs charge membership fees. Club A charges a flat fee of 30 dollars and 10 dollars per month. Club B charges a flat fee of 10 dollars and 15 dollars per month. Write a system of equations to represent this scenario, find the number of months where both clubs cost the same amount, and determine that cost.

Number of months:
Total cost in dollars:

Teacher Copy

Answer Key

  1. Which ordered pair is the solution to the system of equations graphed below?
    Equation 1 is a line that passes through the points (0, 1) and (1, 3).
    Equation 2 is a line that passes through the points (0, 5) and (1, 3).
    A. (0, 1)
    B. (1, 3)
    C. (0, 5)
    D. (2, 5)

  2. Consider the system of equations below:
    y = -2x + 4
    y = -2x - 1
    How many solutions does this system of equations have?
    A. Exactly one solution
    B. No solution
    C. Infinitely many solutions
    D. Exactly two solutions

  3. A rental company offers two plans. Plan A costs a flat fee of 20 dollars plus 2 dollars per hour. Plan B costs a flat fee of 10 dollars plus 4 dollars per hour. Which system of equations represents the cost, y, for x hours of use?
    A. y = 20x + 2 and y = 10x + 4
    B. y = 2x + 20 and y = 4x + 10
    C. y = 22x and y = 14x
    D. y = 2x - 20 and y = 4x - 10

  4. What is the most logical first step to solve the system of equations below using substitution?
    x = y + 3
    2x + 3y = 11
    A. Substitute y + 3 for x in the second equation.
    B. Substitute 2x + 3y for x in the first equation.
    C. Add the two equations together to eliminate x.
    D. Subtract 3 from both sides of the second equation.

  5. Solve the system of equations using substitution by filling in the missing steps:
    y = 3x
    2x + y = 15

Step 1: Substitute 3x for y in the second equation:
2x + = 15

Step 2: Combine like terms:
5x = 15

Step 3: Solve for x:
x =

Step 4: Substitute the value of x back into the first equation to find y:
y = 3 ×
y = 9

  1. Solve the system of equations using elimination by filling in the missing steps:
    4x + y = 13
    2x - y = 5

Step 1: Add the two equations together to eliminate y:
6x =

Step 2: Solve for x:
x =

Step 3: Substitute the value of x back into the first equation to solve for y:
4 × + y = 13
12 + y = 13
y =

  1. Solve the system of equations using any method of your choice. Show your work and check your final answer in both equations.
    x + y = 8
    y = 3x

Final Solution:
x =
y =

  1. Two different fitness clubs charge membership fees. Club A charges a flat fee of 30 dollars and 10 dollars per month. Club B charges a flat fee of 10 dollars and 15 dollars per month. Write a system of equations to represent this scenario, find the number of months where both clubs cost the same amount, and determine that cost.

Number of months:
Total cost in dollars:

Answers and Explanations

  1. B. (1, 3)
    Explanation: The solution to a system of equations graphed on a coordinate plane is the point where the two lines intersect. Since both lines pass through the point (1, 3), this is their point of intersection and the solution to the system.

  2. B. No solution
    Explanation: Both lines have the same slope of -2 but different y-intercepts, meaning they are parallel lines that will never intersect, resulting in zero solutions.

  3. B. y = 2x + 20 and y = 4x + 10
    Explanation: The cost y is represented by multiplying the hourly rate by the number of hours x and adding the flat fee, yielding y = 2x + 20 for Plan A and y = 4x + 10 for Plan B.

  4. A. Substitute y + 3 for x in the second equation.
    Explanation: Since the first equation expresses x in terms of y, substituting this expression into the second equation allows you to solve for a single variable.

  5. Step 1 inline answer: 3x, Step 3 inline answer: 3, Step 4 inline answer: 3
    Explanation: Substituting 3x for y gives 2x + 3x = 15, which simplifies to 5x = 15. Dividing both sides by 5 yields x = 3. Substituting x = 3 back into y = 3x gives y = 3 times 3, which equals 9.

  6. Step 1 inline answer: 18, Step 2 inline answer: 3, Step 3 first inline answer: 3, Step 3 second inline answer: 1
    Explanation: Adding the equations eliminates y, resulting in 6x = 18. Solving for x gives x = 3. Substituting x = 3 into the first equation results in 4 times 3 plus y equals 13, which simplifies to y = 1.

  7. Blank work space: Substitute y = 3x into x + y = 8 to get x + 3x = 8, which simplifies to 4x = 8, meaning x = 2. Then substitute x = 2 into y = 3x to get y = 6. Checking both equations: 2 + 6 = 8 and 6 = 3 multiplied by 2. Final Solution: x = 2, y = 6.
    Explanation: Solving by substitution involves replacing y with 3x in the first equation, solving for x to get 2, and then using that value to determine y is 6.

  8. Blank work space: Let x be the number of months and y be the total cost. The system of equations is y = 10x + 30 and y = 15x + 10. Setting them equal gives 10x + 30 = 15x + 10, which simplifies to 20 = 5x, so x = 4. Substituting back gives y = 10 multiplied by 4 plus 30 = 70. Number of months: 4, Total cost in dollars: 70.
    Explanation: Setting the two cost equations equal to each other allows us to solve for the number of months where the costs are equal, which is 4 months, resulting in a matching cost of 70 dollars.