Substitution is easiest when one equation already has x or y by itself.
Free Interactive Algebra Tool
Systems of Linear Equations Practice
Solve systems of two linear equations step by step using substitution, elimination, or graphing. Practice each method while seeing how both equations work together to produce one solution.
Start practicingNeed Help With Systems of Linear Equations?
Review how substitution, elimination, and graphing can be used to find the ordered pair that satisfies both equations in a system.
Read the Systems of Linear Equations Guide
Solving Method
Choose how you want to solve the system.
Problem 1 of 10
10%
Current system
Identify the equation that already has a variable isolated.
Substitution
Beginner
Original system
y = 2x + 3
3x + y = 13
1
Choose
2
Substitute
3
Solve
4
Back-substitute
5
Solution
Step 1
Rewrite both equations in slope-intercept form
Before graphing, write each equation in the form y = mx + b.
Equation 1
2x + y = 7
Equation 2
x - y = -2
Step 2
Graph the first line
Identify the slope and y-intercept, then use them to construct Equation 1 on the coordinate plane.
Equation 1
y = -2x + 7
Identify the slope and y-intercept
Start at the y-intercept
The y-intercept is the point where the line crosses the y-axis.
First point:
(0, 7)
This point is now marked on the graph.
Use the slope to find a second point
Starting from the y-intercept, use the slope to determine another point on the line.
Equation 1 is graphed.
Both points lie on the line, so the complete
line can now be drawn.
Step 3
Graph the second line
Keep Equation 1 on the graph. Now identify the slope and y-intercept of Equation 2 and construct the second line on the same coordinate plane.
Equation 2
y = x + 2
Identify the slope and y-intercept
Start at the y-intercept
The second line begins with its own y-intercept on the same coordinate plane.
First point:
(0, 2)
This point is now marked on the graph.
Use the slope to find a second point
Starting from the y-intercept, use the slope to determine another point on Equation 2.
Equation 2 is graphed.
Both equations are now visible on the same
coordinate plane.
Step 4
Find the intersection
The solution of a system by graphing is the point where the two lines intersect.
Click the intersection on the graph
Select the point where Equation 1 and Equation 2 cross. The coordinates you select will appear below.
Selected point:
None yet
Or enter the intersection coordinates
Write the point where the two lines meet.
Intersection found.
The point where the two lines meet is the
solution of the system.
Step 5
Verify the solution
The intersection should satisfy both original equations. Check the ordered pair in each equation.
Intersection
(2, 3)
Equation 1
x + y = 5
2 + 3 = 5
Equation 2
2x - y = 1
2(2) - 3 = 1
Does the intersection satisfy both equations?
Graphing Lab
Coordinate plane
Step 1
Which equation already has a variable isolated?
For substitution, look for an equation where either x or y is already by itself.
Step 2
Substitute the isolated expression
Replace y in the other equation with the expression that y equals.
3x + ? = 13
Step 3
Solve the resulting equation
After substitution, the system has become a one-variable equation. Solve it to find the first coordinate.
3x + (2x + 3) = 13
Your work
0 steps completed
- Start 3x + (2x + 3) = 13
Step 4
Substitute back to find the other variable
Substitute the value of x into one of the original equations and solve for y.
y = 2(2) + 3
Step 5
Write the solution as an ordered pair
The solution to a system is written as (x, y).
Step 1
Which variable should we eliminate?
Compare the x-coefficients and y-coefficients. Choose the variable that can be eliminated most efficiently.
Step 2
Prepare the equations for elimination
Make the chosen variable cancel, then combine the equations.
Choose a multiplier for each equation
Equations to combine
2x + 3y = 12
4x - 3y = 6
How should the equations be combined?
Step 3
Solve for the remaining variable
One variable has been eliminated. Solve the resulting one-variable equation one algebra step at a time.
6x = 18
Your work
0 steps completed
- Start 6x = 18
Step 4
Substitute back to find the other variable
Substitute the value you found into either original equation.
2(3) + 3y = 12
Step 5
Write the solution as an ordered pair
Combine the values you found and write the system solution in the form (x, y).
Start by identifying the equation that already has a variable isolated.
Hint
Look for the equation where x or y is already alone on one side.
Systems of equations reference
Replace the isolated variable in the other equation with the expression it equals.
Simplify and solve the resulting one-variable equation.
Use the value you found to calculate the other variable.
Always write the final system solution in the order (x, y).
A true system solution must make both original equations true.
System solved
Solution: (2, 7)
This ordered pair makes both original equations true.
Systems Solved
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Correct Steps
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Incorrect Attempts
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Current Streak
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Practice Round Complete
Great work!
You completed all 10 systems of linear equations.
Systems Solved
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Step Accuracy
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Substitute one expression
When one variable is isolated, replace that variable in the other equation with the expression it equals.
Solve one variable first
After substitution, the two-equation system becomes a single equation containing only one variable.
Finish with an ordered pair
Substitute back to find the second variable, then write the final solution as an ordered pair in the form (x, y).
Example
What does substitution look like?
Suppose the system is:
y = 2x + 3
3x + y = 13
1
Identify the isolated variable
y = 2x + 3
2
Substitute for y
3x + (2x + 3) = 13
3
Solve for x
5x + 3 = 13
5x = 10
x = 2
4
Substitute back
y = 2(2) + 3
y = 7
5
Write the solution
(2, 7)
The ordered pair (2, 7) satisfies both original equations, so it is the solution to the system.
Practice Solving Systems of Linear Equations Step by Step
A system of linear equations contains two or more equations whose variables must satisfy every equation at the same time. For a system containing two variables, the solution is usually written as an ordered pair in the form (x, y).
This interactive practice tool helps students learn the major methods used to solve systems of linear equations: substitution, elimination, and graphing.
Solve Systems Using Substitution
The substitution method works by solving one equation for a variable and then replacing that variable in the other equation with an equivalent expression. This creates a one-variable equation that can be solved using ordinary algebra.
Solve Systems Using Elimination
The elimination method combines two equations so that one variable cancels. Some systems can be added or subtracted immediately, while others require multiplying one or both equations first.
Solve Systems by Graphing
When a system is solved by graphing, each equation is represented by a line. The point where the two lines intersect is the ordered pair that satisfies both equations.
Understanding the Solution
The solution to a system is not simply an x-value or a y-value. It is the combination of values that makes both equations true at the same time. For two-variable systems, this is represented by an ordered pair.
Three Difficulty Levels
Beginner substitution problems begin with a variable already isolated and emphasize integer solutions. Intermediate problems introduce negative values and systems that may require an additional algebra step before substitution. Advanced problems can include more complex coefficients and fractional values.
Free Systems of Equations Practice on Any Device
The tool is designed for desktop computers, tablets, and phones and can be used during tutoring sessions, homework practice, classroom review, or independent study.