Systems of Equations

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A system of equations is two or more equations that use the same variables. Its solution is the point that makes every equation true at the same time. When you graph a system of two linear equations, the solution is the point where the two lines cross.

Before this lesson: You should know how to graph a line in slope-intercept form and what a solution to a two-variable linear equation is.

What Is a System of Equations?

One linear equation with x and y has many solutions. Every point on its line works. A system asks a harder question: which point works for both equations at once?

Here is a system of two equations. The brace shows that they go together.

\begin{cases}
y = x + 1 \\
y = -x + 5
\end{cases}

The point (2, 3) is the solution. It is on both lines, so it makes both equations true.

\begin{aligned}
3 &= 2 + 1 && \checkmark \\
3 &= -2 + 5 && \checkmark
\end{aligned}
The lines y = x + 1 and y = −x + 5 cross at the point (2, 3), which is the solution of the system.
The lines y = x + 1 and y = −x + 5 cross at (2, 3), the solution of the system.

How to Check a Solution

  1. Put the x-value and y-value into the first equation.
  2. Put the same values into the second equation.
  3. If both equations are true, the point is a solution. If either one is false, it is not.

How to Solve a System by Graphing

  1. Write each equation in slope-intercept form, y = mx + b.
  2. Graph both lines on the same grid.
  3. Find the point where the lines cross.
  4. Check that point in both equations.

Worked Examples

Example 1: Check a solution that works

Is (3, −1) a solution of this system?

\begin{cases}
x + y = 2 \\
2x - y = 7
\end{cases}

Step 1. First equation: 3 + (-1) = 2. True.

Step 2. Second equation: 2(3) - (-1) = 6 + 1 = 7. True.

Answer: Yes. (3, −1) makes both equations true, so it is the solution.

Example 2: Check a point that doesn’t work

Is (1, 4) a solution of y = 2x + 2 and y = -x + 3?

Step 1. First equation: 2(1) + 2 = 4. True.

Step 2. Second equation: -1 + 3 = 2, not 4. False.

Answer: No. The point is on the first line but not the second, so it is not a solution of the system.

Example 3: Solve by graphing

Solve the system y = x + 1 and y = -x + 5 by graphing.

Step 1. Graph y = x + 1: start at (0, 1), then go up 1 and right 1.

Step 2. Graph y = -x + 5: start at (0, 5), then go down 1 and right 1.

Step 3. The lines cross at (2, 3), as shown in the graph above.

Answer: (2, 3). The check above shows it works in both equations.

Example 4: Rewrite first, then graph

Solve the system 2x + y = 4 and x - y = -1 by graphing.

Step 1. Solve each equation for y.

\begin{aligned}
2x + y = 4 \quad &\Rightarrow \quad y = -2x + 4 \\
x - y = -1 \quad &\Rightarrow \quad y = x + 1
\end{aligned}

Step 2. Graph both lines. The first starts at (0, 4) with slope −2. The second starts at (0, 1) with slope 1.

The lines y = −2x + 4 and y = x + 1 cross at the point (1, 2).
The lines y = −2x + 4 and y = x + 1 cross at (1, 2).

Step 3. The lines cross at (1, 2).

Check: 2(1) + 2 = 4 and 1 - 2 = -1. Both are true.

Answer: (1, 2)

Example 5: Word problem

Streaming Plan A costs $20 to join plus $5 per month. Plan B has no fee and costs $10 per month. After how many months do the plans cost the same?

Step 1. Let m be the number of months and C the total cost. Write an equation for each plan.

\begin{cases}
C = 5m + 20 & \text{Plan A} \\
C = 10m & \text{Plan B}
\end{cases}

Step 2. Graph both lines. Plan A starts at (0, 20) and rises $5 per month. Plan B starts at (0, 0) and rises $10 per month.

Step 3. The lines cross at (4, 40).

Check: Plan A: 5(4) + 20 = 40. Plan B: 10(4) = 40.

Answer: After 4 months, both plans cost $40. Before that Plan B is cheaper, and after that Plan A is cheaper.

When Graphing Isn’t Enough

Graphing shows you what a solution means, but it isn’t always exact. If the lines cross at a point like \left(\frac{7}{3}, \frac{5}{2}\right), you can only estimate it from a graph.

For exact answers, use algebra. The next two lessons show how: the Substitution Method and the Elimination Method.

Sometimes the lines never cross, or they are the same line. You’ll learn those cases in Number of Solutions.

Common Mistakes

MistakeWhat it looks likeHow to fix it
Checking only one equation(1, 4) works in y = 2x + 2, so you call it the solutionA solution must work in every equation in the system.
Switching x and yReading the crossing point as (3, 2) instead of (2, 3)Write the x-value first: (x, y).
Graphing before solving for yGraphing 2x + y = 4 with slope 2Solve for y first: y = -2x + 4 has slope −2.
Trusting a rough sketchGuessing (2, 3) when the lines really cross at (2.2, 3.1)Use a ruler and graph paper, and always check the point in both equations.

Practice Problems

For problems 1–3, decide if the point is a solution of the system.

  1. (2, 5); y = 3x - 1 and y = x + 3
  2. (−1, 2); x + y = 1 and 3x - y = -4
  3. (0, −3); y = 4x - 3 and 2x + y = -3

For problems 4–8, solve the system by graphing. Check your answer.

  1. y = x - 2 and y = -x + 4
  2. y = 2x + 1 and y = -x - 2
  3. y = \frac{1}{2}x + 1 and y = -x + 4
  4. x + y = 5 and x - y = 1
  5. y = 3 and x = -2
  6. Ana has $10 and saves $5 each week. Ben has $30 and saves $3 each week. After how many weeks will they have the same amount? How much will each have?
Show answers
  1. Yes. 3(2) - 1 = 5 and 2 + 3 = 5. Both are true.
  2. No. -1 + 2 = 1 is true, but 3(-1) - 2 = -5, not −4.
  3. Yes. 4(0) - 3 = -3 and 2(0) + (-3) = -3. Both are true.
  4. (3, 1). Check: 3 - 2 = 1 and -3 + 4 = 1.
  5. (−1, −1). Check: 2(-1) + 1 = -1 and -(-1) - 2 = -1.
  6. (2, 2). Check: \frac{1}{2}(2) + 1 = 2 and -2 + 4 = 2.
  7. Rewrite as y = -x + 5 and y = x - 1. They cross at (3, 2). Check: 3 + 2 = 5 and 3 - 2 = 1.
  8. (−2, 3). The horizontal line y = 3 and the vertical line x = -2 cross there.
  9. Ana: A = 5w + 10. Ben: A = 3w + 30. The lines cross at (10, 60). After 10 weeks, each has $60.

Systems of Equations FAQ

What is a system of equations?

A system of equations is a group of two or more equations with the same variables. You solve the equations together to find values that make all of them true.

What does the solution of a system mean?

The solution is the point that lies on every line in the system. On a graph, it is where the lines cross.

What are the three ways to solve a system of equations?

The three main methods are graphing, substitution and elimination. Graphing helps you see the answer, while substitution and elimination give exact answers.

Can a system of equations have no solution?

Yes. If the lines are parallel, they never cross, so there is no solution. See Number of Solutions.

Summary

  • A system of equations is two or more equations with the same variables.
  • The solution makes every equation true at the same time.
  • On a graph, the solution is where the lines cross.
  • To solve by graphing, write each equation as y = mx + b, graph both lines and find where they cross.
  • Always check the point in every equation.

Previous lesson: Arithmetic Sequences

Next lesson: Substitution Method

Back to the Algebra 1 Index

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