Equation Solver
To solve an equation is to find the value that makes both sides equal. This solver handles the two forms students meet first: a single linear equation with x on both sides, and a pair of equations with two unknowns. It also tells you when there is no solution or infinitely many, and it substitutes its answer back in so you can see the check work.
How to use the equation solver
- Pick the type. For a linear equation enter a, b, c and d for ax + b = cx + d. For a system enter a, b, c for the first equation (ax + by = c) and d, e, f for the second.
- Read the solution in the highlighted rows and follow the steps below them.
- Compare the two check rows. If they match the right-hand side, the answer is verified.
Formula
Linear: collect x-terms on one side: (a − c)x = d − b, so x = (d − b) ÷ (a − c).
System (Cramer’s rule): with determinant D = ae − bd, x = (ce − bf) ÷ D and y = (af − cd) ÷ D. If D = 0 the lines are parallel (no solution) or identical (infinitely many).
Worked examples
A linear equation
3x + 5 = x + 11. Subtract x and 5: 2x = 6, so x = 3. Check: 3(3) + 5 = 14 and 3 + 11 = 14.
A system
2x + y = 8 and x − y = 1. Adding the equations gives 3x = 9, so x = 3 and y = 2. Check: 2(3) + 2 = 8 and 3 − 2 = 1.
No solution
2x + 1 = 2x + 5 simplifies to 1 = 5, which is never true, so there is no solution.
How to solve equations by hand
The balance idea
An equation is a balanced scale. Whatever you do to one side, you must do to the other. To solve 3x + 5 = x + 11, subtract x from both sides to get 2x + 5 = 11, subtract 5 to get 2x = 6, then divide by 2 to get x = 3. Each step keeps the scale level.
Three possible outcomes
| Result | What it looks like | Meaning |
|---|---|---|
| One solution | x = 3 | The lines cross once |
| No solution | 1 = 5 | Parallel lines never meet |
| Infinitely many | 4 = 4 | Both sides are the same line |
Solving systems: substitution and elimination
For two unknowns you can substitute one equation into the other, or add and subtract the equations to eliminate a variable. In 2x + y = 8 and x − y = 1, adding the equations removes y and gives 3x = 9. This tool uses Cramer’s rule, which gives the same answer directly from the coefficients.
Checking your work
Always substitute your answer back into the original equation. If both sides agree, you are done. It takes ten seconds and catches almost every arithmetic slip.
Practice problems
Try these yourself first, then check your answer. The answers come from the calculator above.
- Solve 4x − 3 = x + 12.
Show answer
x: 5
- Solve the system x + y = 10 and x − y = 2.
Show answer
x: 6 · y: 4
Common mistakes to avoid
- Forgetting to change the sign when moving a term across the equals sign.
- Dividing by zero when a = c. That case means either no solution or every value works.
- Mixing up which coefficients belong to which equation in a system.
Frequently asked questions
How do I solve for x?
Get all the x-terms on one side and the numbers on the other, then divide by the coefficient of x.
What does “no solution” mean?
The equation simplifies to something false, such as 1 = 5. Graphically, the two sides are parallel lines that never meet.
What is a system of equations?
Two or more equations that share unknowns. The solution must satisfy all of them at once.
Can it solve quadratics?
Use the quadratic equation solver for equations with an x² term.
Reviewed September 26, 2026 by the SolveCalcPro editorial team. Found a mistake? Tell us; see our editorial policy.