What Is a Solution in Math? Solution Math Examples

Q: How can an equation have no solution?

When solving leads to a false statement such as 6 = 1. No value of x can make it true, so the solution set is empty.

Q: What does “all real numbers” mean as an answer?

That every value of x works: solving gives a true statement such as -3 = -3. The equation is an identity, and it has infinitely many solutions.

What is a solution in math? Examples with one, none or many

A solution is a value that makes an equation true. How to check one, and examples with one solution, no solution, all real numbers and two solutions.

2 min readUpdated 2 November 2026By Sumzo team (AI-assisted)Reviewed by Sumzo AI (owner-delegated review)

A solution in math is a value that makes an equation true. For $x + 3 = 10$ the solution is $x = 7$ , because $7 + 3 = 10$ . The solution math examples below show how to check a solution, and how an equation can have one solution, no solution, or infinitely many.

How to check a solution

Substitute the value into the original equation. If both sides come out equal, it is a solution; if not, it isn’t.

Worked example

Easy: is $x = 4$ a solution?

Is $x = 4$ a solution of $3x - 5 = 7$ ?

Show the working

Substitute $x = 4$ into the left side. $3(4) - 5 = 12 - 5 = 7$
The left side equals the right side, 7. $7 = 7 \checkmark$

Answer: Yes, $x = 4$ is a solution.

Checked by Sumzo’s math engine

One solution, no solution or all real numbers

Worked example

Medium: one solution

Solve $2(x + 3) = 4x - 2$ .

Show the working

Expand the bracket. $2x + 6 = 4x - 2$
Collect $x$ on one side. $8 = 2x \Rightarrow x = 4$
Check: $2(7) = 14$ and $4(4) - 2 = 14$ ✓.

Answer: One solution: $x = 4$ .

Checked by Sumzo’s math engine

No solution: $2(x + 3) = 2x + 1$ becomes $2x + 6 = 2x + 1$ , so $6 = 1$ — false for every $x$ . The solution set is empty.
All real numbers: $3(x - 1) = 3x - 3$ becomes $3x - 3 = 3x - 3$ — true for every $x$ . Every real number is a solution (infinitely many solutions).

All real numbers vs no solution

Both happen when the $x$ terms cancel. If what is left is true (like $-3 = -3$ ), the answer is all real numbers; if it is false (like $6 = 1$ ), there is no solution.

Equations with more than one solution

Worked example

Medium: two solutions

Solve $x^2 = 9$ .

Show the working

Both $3^2$ and $(-3)^2$ equal 9. $x = 3 \text{ or } x = -3$

Answer: $x = 3$ or $x = -3$

Checked by Sumzo’s math engine

Worked example

Hard: the solution of a system is a point

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

Show the working

Add the equations to eliminate $y$ . $2x = 6 \Rightarrow x = 3$
Substitute back. $3 + y = 5 \Rightarrow y = 2$
Check in both equations: $3 + 2 = 5$ ✓ and $3 - 2 = 1$ ✓.

Answer: The solution is the point $(3, 2)$ — where the two lines cross.

Checked by Sumzo’s math engine

How to graph no solution on a number line

The solutions of an inequality are shown by shading a number line. If there is no solution — for example $x < 2$ and $x > 5$ at the same time — nothing is shaded. If every real number is a solution, the whole line is shaded, with arrows at both ends.

Common mistakes

Checking in a rearranged equation instead of the original one.
Writing $x = 0$ when the answer is “no solution” — 0 is a number, “no solution” means none works.
Stopping after one solution when there are two, as with $x^2 = 9$ .
Forgetting that the solution of a system needs both values, $x$ and $y$ .

Frequently asked questions