Linear Equation Solver
Solve a linear equation written as ax + b = c for x, with each inverse operation shown and the answer checked by substitution.
Calculator
- Isolates x by inverse operations
- One / none / infinite solutions
- Exact fraction form
- Substitution check
Result & calculation receipt
- Method
- Isolate x by inverse operations
- Formula
ax + b = c ⟹ x = (c − b) / a- Substitution
- x = (22 − 7) / 3 = 15 / 3 = 5
- Steps
- Subtract b: 3·x = 22 − 7 = 15.
- Divide by a: x = 15 ÷ 3 = 5.
- Verification
- Substitute back: 3·(5) + 7 = 22 = 22. ✓
- Exact result
- 5
- Decimal result
- 5
- Engine
- math-solvers v1.4.0
How this calculator works
The calculator isolates x by undoing the operations around it: subtract the constant b from both sides, then divide both sides by the coefficient a. It also handles the special cases where a is zero — the equation has no solution if the constants disagree, or infinitely many if they match — and gives the exact fractional answer when x is not a whole number.
Worked example
For 3x + 7 = 22, subtracting 7 gives 3x = 15, and dividing by 3 gives x = 5. Substituting back, 3 × 5 + 7 = 22, which matches. If the equation were 0x + 5 = 9, no value of x can make 5 equal 9, so there is no solution; if it were 0x + 5 = 5, every x works.
Key insight
Every linear equation in one variable rearranges to ax + b = c, so once you can solve that form you can solve them all. The three outcomes — one solution, none, or infinitely many — depend entirely on whether a (and then b versus c) is zero.
Frequently asked questions
How do I solve ax + b = c for x?
Undo the operations around x: subtract b from both sides to get ax = c − b, then divide both sides by a to get x = (c − b)/a. The calculator shows both steps.
When does a linear equation have no solution?
When the coefficient a is zero and the remaining constants disagree, such as 0x + 5 = 9. There is no x that makes a false numerical statement true, so no solution exists.
When does it have infinitely many solutions?
When a is zero and the constants agree, such as 0x + 5 = 5. The equation is true for every value of x, so any number is a solution.
Can the answer be a fraction?
Yes. When (c − b) is not divisible by a, the calculator gives the exact fraction as well as the decimal, for example x = 7/3.
How is the solution verified?
By substitution: the calculator puts x back into ax + b and confirms the result equals c, and shows that check in the receipt.
Methodology & verification
A linear equation is normalized to ax + b = c and solved by inverse operations: x = (c − b)/a. When a = 0 the calculator distinguishes no solution (b ≠ c) from infinitely many (b = c) rather than dividing by zero. The answer is given as an exact fraction (when non-integer) and a decimal, and is verified by substituting x back into the left-hand side and confirming it equals c.
Engine: math-solvers v1.4.0 · Last reviewed: · Every result is computed client-side and never sent to a server. See the verification methodology and the scientific calculator validation.