What does no solution mean for a system of equations?

The two lines are parallel, so no point sits on both of them. Solving gives a false statement such as 1 = −3.

How do you know if a system has infinitely many solutions?

The two equations describe the same line. Solving gives a statement that is always true, such as 0 = 0, and every point on the line is a solution.

Can two lines with the same slope have a solution?

Only if they are the same line. With the same slope and different intercepts they are parallel and never meet, so there is no solution.

How can you tell without graphing?

Write both equations as y = mx + b. Different slopes mean one solution. The same slope with different b means none. The same slope and the same b means infinitely many.

Part 17 of 20 · EquationsSee the whole course

No solution or infinitely many solutions: how to tell

By The Pip team · October 23, 2026 · 6 min read

Short answer

Lines with the same slope and different intercepts never meet: no solution. Lines that are really the same line meet everywhere: infinitely many solutions. Any other pair of lines crosses exactly once: one solution.

Three possible outcomes

A system of two linear equations can end in exactly one of three ways:

One solution. The slopes are different, so the lines cross once.
No solution. The slopes are equal and the intercepts differ. The lines are parallel.
Infinitely many solutions. The slopes and the intercepts are both equal. It is one line written twice.

To tell which one without graphing, put both equations in y = mx + b form and compare the slopes first, then the intercepts.

Example 1: no solution

Solve y = 2x + 1 and y = 2x − 3. Both slopes are 2, but the intercepts differ. Set the two right-hand sides equal:

2x + 1 = 2x − 3
Subtract 2x from both sides: 1 = −3
That is false, whatever x is, so there is no solution.

Example 2: infinitely many solutions

Solve y = 2x + 1 and 2y = 4x + 2. The second equation looks different, but divide it by 2:

2y = 4x + 2 becomes y = 2x + 1, the same line.
Check by substituting: 2(2x + 1) = 4x + 2, which is 4x + 2 = 4x + 2.
Subtract 4x + 2 from both sides: 0 = 0. That is always true, so every point on the line is a solution.

Example 3: one solution

y = 2x + 1 and y = −x + 7
2x + 1 = −x + 7
3x = 6, so x = 2, and y = 2(2) + 1 = 5
The slopes differ, so there is exactly one solution: (2, 5).

Try all three below. Set line 2 to slope 2 and slide b: only one value of b gives the same line.

No solution

y = 2x + 1
y = 2x − 3

Line 2 slope m 2

Line 2 starts at b −3

2x + 1 = 2x − 3
1 = −3: never true

Same slope, different b: the lines are parallel and never meet. The algebra ends in a false statement, so there is no solution.

What the algebra is telling you

A false statement such as 1 = −3: no solution.
A statement that is always true such as 0 = 0: infinitely many solutions.
An answer like x = 2: one solution. Find y to finish, as in substitution.

Watch it explained in under a minute

The same idea, narrated step by step. Square version for sharing: download the 1:1 video.

The four mistakes to avoid

Reading 0 = 0 as x = 0. It means every value of x works, not just zero.
Forcing an answer out of a false statement like 1 = −3. “No solution” is a real answer, not a slip.
Comparing slopes before rearranging. In 2y = 4x + 2 the slope is 2 (divide by 2), not 4.
Thinking the same slope always means no solution. With the same slope and the same b, it is the same line.

Try it yourself

How many solutions? Tap a question to see the answer.

1. y = 3x + 2 and y = 3x − 4

Same slope (3), different b. The lines are parallel: no solution.

2. y = x + 1 and 2y = 2x + 2

Divide the second by 2: y = x + 1. It is the same line: infinitely many solutions.

3. y = 4x and y = 2x + 6

The slopes differ, so there is one solution. 4x = 2x + 6 gives x = 3, and y = 12: (3, 12).

Where this fits in

Systems of equations are everywhere in real choices, like comparing two phone plans. Next in this course: the break-even point.