Number of Solutions for Systems of Equations
A system of two linear equations has one solution, no solution or infinitely many solutions. Lines with different slopes cross once, so there is one solution. Parallel lines never cross, so there is no solution. If both equations make the same line, every point on it is a solution.
Before this lesson: You should know how to read slope and y-intercept in slope-intercept form and how to solve a system with the elimination method.
The Three Cases
Two lines on a graph can only do three things: cross once, never cross, or lie on top of each other.
| Number of solutions | What the graph shows | Slopes and y-intercepts | What algebra gives you |
|---|---|---|---|
| One solution | The lines cross at one point | Different slopes | One value for x and one for y |
| No solution | The lines are parallel | Same slope, different y-intercepts | A false statement, like 0 = 4 |
| Infinitely many solutions | The lines are the same line | Same slope and same y-intercept | A true statement, like 0 = 0 |
Your textbook may use these words too:
- A consistent system has at least one solution.
- An inconsistent system has no solution.
- A dependent system has infinitely many solutions, because the equations describe the same line.
Method 1: Compare Slopes and Y-Intercepts
- Write both equations in the form y = mx + b.
- If the slopes are different, there is one solution.
- If the slopes are the same, compare the y-intercepts. Different y-intercepts mean no solution. The same y-intercept means infinitely many solutions.
This method tells you how many solutions there are without solving the system.
Method 2: Solve and Read the Result
Solve the system with substitution or elimination.
- If you get values like x = 2 and y = 5, there is one solution.
- If both variables cancel and you get a false statement, there is no solution.
- If both variables cancel and you get a true statement, there are infinitely many solutions.
Worked Examples
Example 1: One solution
How many solutions does the system y = 2x + 1 and y = -x + 4 have?
Step 1. Both equations are already in slope-intercept form.
Step 2. The slopes are 2 and −1. They are different, so the lines cross once.
Answer: One solution. Solving gives (1, 3).
Example 2: No solution
How many solutions does the system y = 3x - 2 and y = 3x + 5 have?
Step 1. Both slopes are 3, so the lines are parallel or the same line.
Step 2. The y-intercepts are −2 and 5. They are different, so the lines are parallel.
Answer: No solution. Parallel lines never cross.
Example 3: Infinitely many solutions
How many solutions does the system 2x + y = 6 and 4x + 2y = 12 have?
Step 1. Solve each equation for y.
\begin{aligned}
2x + y = 6 \quad &\Rightarrow \quad y = -2x + 6 \\
4x + 2y = 12 \quad &\Rightarrow \quad y = -2x + 6
\end{aligned}Step 2. The equations are the same line. The second equation is just the first one multiplied by 2.
Answer: Infinitely many solutions. Every point on y = -2x + 6 works, such as (0, 6), (1, 4) and (3, 0).
Example 4: Use elimination
How many solutions does the system x + y = 4 and 2x + 2y = 10 have?
Step 1. Multiply the first equation by 2: 2x + 2y = 8.
Step 2. Subtract it from the second equation.
\begin{array}{rrcl}
& 2x + 2y & = & 10 \\
- & (2x + 2y) & = & 8 \\
\hline
& 0 & = & 2
\end{array}Step 3. Both variables cancel, and 0 = 2 is false.
Answer: No solution. In slope-intercept form, the lines are y = -x + 4 and y = -x + 5. They have the same slope but different y-intercepts.
Example 5: Word problem
Jake has $50 and saves $10 each week. Mia has $80 and also saves $10 each week. Will they ever have the same amount of money?
Step 1. Let w be the number of weeks and A the amount saved.
\begin{cases}
A = 10w + 50 & \text{Jake} \\
A = 10w + 80 & \text{Mia}
\end{cases}Step 2. Both slopes are 10, but the starting amounts are different.
Answer: No. The lines are parallel, so the system has no solution. Mia stays $30 ahead every week.
Common Mistakes
| Mistake | What it looks like | How to fix it |
|---|---|---|
| Checking only the slopes | Same slope, so “infinitely many” | Same slope can also mean parallel. Compare the y-intercepts too. |
| Reading 0 = 0 as x = 0 | “The answer is (0, 0).” | 0 = 0 means every point on the line is a solution. |
| Comparing before solving for y | 2x + y = 3 and 2x - y = 1 called parallel | Solve for y first. The slopes are −2 and 2, so there is one solution. |
| Thinking “no solution” means a mistake | Starting over after getting 0 = 4 | A false statement is a real answer. It means the lines are parallel. |
Practice Problems
How many solutions does each system have? For problems 7 and 8, solve with elimination.
- y = 4x - 1 and y = 4x + 6
- y = -2x + 3 and y = x + 3
- y = \frac{1}{2}x + 2 and 2y = x + 4
- x + y = 5 and x + y = 9
- 3x - y = 2 and 6x - 2y = 4
- 2x + 3y = 6 and 3x - 2y = 6
- x - 2y = 3 and -2x + 4y = -6
- 3x + y = 7 and 6x + 2y = 10
- What value of k makes the system y = kx + 3 and y = 4x + 1 have no solution?
- Two pools each hold 200 gallons. Both are filled at 25 gallons per minute. How many times do they hold the same amount of water?
Show answers
- No solution. Same slope (4), different y-intercepts (−1 and 6).
- One solution. The slopes are −2 and 1. The lines cross at (0, 3).
- Infinitely many. The second equation becomes y = \frac{1}{2}x + 2, the same line.
- No solution. x + y can’t equal 5 and 9 at the same time. Both lines have slope −1, with different y-intercepts.
- Infinitely many. Both become y = 3x - 2.
- One solution. The slopes are -\frac{2}{3} and \frac{3}{2}, which are different.
- Multiply the first equation by 2: 2x - 4y = 6. Add it to the second: 0 = 0. Infinitely many solutions.
- Multiply the first equation by 2: 6x + 2y = 14. Subtract the second: 0 = 4, which is false. No solution.
- k = 4. Then both lines have slope 4 with different y-intercepts (3 and 1), so they are parallel.
- Infinitely many. Both pools follow G = 25t + 200, so they always hold the same amount.
Number of Solutions FAQ
How do you know if a system has no solution?
A system has no solution when the lines have the same slope but different y-intercepts. When you solve it, the variables cancel and you get a false statement like 0 = 4.
What does infinitely many solutions mean?
It means both equations describe the same line. Every point on that line makes both equations true.
Can a system of two linear equations have exactly two solutions?
No. Two straight lines can cross at most once, so the only choices are one solution, no solution or infinitely many.
What is an inconsistent system?
An inconsistent system is a system with no solution. Its lines are parallel.
Summary
- A system of two linear equations has one solution, no solution or infinitely many solutions.
- Different slopes: one solution.
- Same slope and different y-intercepts: no solution (parallel lines).
- Same slope and same y-intercept: infinitely many solutions (same line).
- When solving, a false statement means no solution, and 0 = 0 means infinitely many.
Previous lesson: Elimination Method
Next lesson: Systems of Equations Word Problems
Later in this unit: Graphing Linear Inequalities
Back to the Algebra 1 Index