Number of Solutions of 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.

Extra practice: The Algebra 1 Unit 5 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.

The Three Cases

Two lines on a graph can only do three things: cross once, never cross, or lie on top of each other.

Three small graphs. Left: two lines that cross once, so one solution. Middle: two parallel lines that never cross, so no solution. Right: two equations that make the same line, so infinitely many solutions. — Lines that cross have one solution. Parallel lines have none. The same line has infinitely many.

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, $y = mx + b$ . Read the slopes: 2 and $-1$ .

Step 2. The slopes are different, so the lines cross exactly once.

2x + 1 = -x + 4

Step 3. To find the point, set the two expressions for $y$ equal (see substitution method).

3x + 1 = 4

Step 4. Add $x$ to both sides.

3x = 3

Step 5. Subtract 1 from both sides.

x = 1

Step 6. Divide both sides by 3.

y = 2(1) + 1 = 2 + 1 = 3

Step 7. Substitute $x = 1$ into $y = 2x + 1$ .

Answer: One solution, the point (1, 3).

Example 2: No solution

How many solutions does the system $y = 3x - 2$ and $y = 3x + 5$ have?

Step 1. Read the slopes. Both slopes are 3, so the lines are parallel or the same line.

Step 2. Read the y-intercepts: $-2$ 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?

y = -2x + 6

Step 1. Solve the first equation for $y$ . Subtract $2x$ from both sides. (See literal equations.)

2y = -4x + 12

Step 2. Solve the second equation for $y$ . Subtract $4x$ from both sides.

y = -2x + 6

Step 3. Divide both sides by 2.

Step 4. The two equations are the same line. The second equation is 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?

2x + 2y = 8

Step 1. Multiply every term of the first equation by 2 (see elimination method).

\begin{array}{rrcl} & 2x + 2y & = & 10 \\ - & (2x + 2y) & = & 8 \\ \hline & 0 & = & 2 \end{array}

Step 2. Subtract it from the second equation.

Step 3. Both variables cancel, and $0 = 2$ is false. No pair of numbers can make it true.

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?

\begin{cases} A = 10w + 50 & \text{Jake} \\ A = 10w + 80 & \text{Mia} \end{cases}

Step 1. Let $w$ be the number of weeks and $A$ the amount saved. Write an equation for each person.

Step 2. Compare the slopes. Both are 10, so the lines are parallel or the same line.

Step 3. Compare the starting amounts, the y-intercepts: 50 and 80. They are different, so the lines are parallel.

10w + 50 = 10w + 80

Step 4. Set the amounts equal to confirm.

50 = 80

Step 5. Subtract $10w$ from both sides.

Step 6. $50 = 80$ is false, so there is no solution.

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

Read the slopes: both are 4
Read the y-intercepts: −1 and 6. They are different
No solution. The lines are parallel.

Read the slopes: −2 and 1. They are different, so there is one solution

-2x + 3 = x + 3

Set the expressions equal.

-3x + 3 = 3

Subtract x from both sides.

-3x = 0

Subtract 3 from both sides.

x = 0

Divide both sides by −3.

y = 0 + 3 = 3

Substitute into y = x + 3.

One solution, (0, 3).

y = \frac{1}{2}x + 2

Solve the second equation for y. Divide both sides by 2.

That is the same as the first equation
Infinitely many solutions.

y = -x + 5

Solve the first equation for y. Subtract x from both sides.

y = -x + 9

Solve the second equation for y. Subtract x from both sides.

Both slopes are −1, and the y-intercepts are 5 and 9
No solution. The lines are parallel.

-y = -3x + 2

Solve the first equation for y. Subtract 3x from both sides.

y = 3x - 2

Divide both sides by −1.

-2y = -6x + 4

Solve the second equation for y. Subtract 6x from both sides.

y = 3x - 2

Divide both sides by −2.

Both are the same line, so there are infinitely many solutions.

3y = -2x + 6

Solve the first equation for y. Subtract 2x from both sides.

y = -\frac{2}{3}x + 2

Divide both sides by 3.

-2y = -3x + 6

Solve the second equation for y. Subtract 3x from both sides.

y = \frac{3}{2}x - 3

Divide both sides by −2.

The slopes are −2/3 and 3/2. They are different
One solution.

2x - 4y = 6

Multiply every term of the first equation by 2.

0 = 0

Add it to the second equation. Every term cancels.

0 = 0 is always true
Infinitely many solutions.

6x + 2y = 14

Multiply every term of the first equation by 2.

0 = 4

Subtract the second equation. Every variable cancels.

0 = 4 is false
No solution.

For no solution, the lines must be parallel: same slope, different y-intercepts

k = 4

The second line has slope 4, so set the first slope equal to it.

The y-intercepts are 3 and 1, which are different
k = 4.

10.

G = 25t + 200

Let t be minutes and G gallons. Write the first pool’s equation.

G = 25t + 200

Write the second pool’s equation.

25(2) + 200 = 50 + 200 = 250

The equations are the same line, so the amounts are equal at every time. For example, at 2 minutes.

Infinitely many. They always hold the same amount.

Want more practice? The Algebra 1 Unit 5 Practice Pack has 55 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

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