Equation Solver
Solve linear equations of the form ax + b = cx + d with full isolating steps and a check that the solution works.
Equation Solver — input your values
What does this math calculator do?
The equation solver handles linear equations of the form ax + b = cx + d, the standard algebra problem. It walks through the classic isolation method: gather every x term on one side, move the constants to the other, and divide by the coefficient of x to reveal the solution.
The calculator then verifies the answer by substituting x back into both sides of the original equation — if the two sides match, the solution is confirmed. Special cases are handled explicitly: when the sides are parallel there is no solution, and when both sides are identical there are infinitely many.
Formula and variables
If a = c and b = d the equation is an identity with infinitely many solutions; if a = c but b ≠ d it has no solution.
| Symbol | Meaning |
|---|---|
| a, c | Coefficients of x on each side |
| b, d | Constants on each side |
| x | The variable to solve for |
Worked example
Solve 3x + 5 = 2x + 9.
- Subtract 2x from both sides: 3x − 2x + 5 = 9, so x + 5 = 9.
- Subtract 5 from both sides: x = 4.
- Check: 3(4) + 5 = 17 and 2(4) + 9 = 17.
Answer: x = 4.
Tips and common mistakes
Tips
- Whatever you do to one side of an equation, you must do to the other — that is the golden rule of algebra.
- Combine like terms before isolating: gather all the x terms first, then the constants.
- Always check your answer by substituting it back into the original equation.
- If your coefficient of x is a fraction, multiply both sides by the denominator to clear it early.
Common mistakes to avoid
- Subtracting a term from only one side, which breaks the balance of the equation.
- Forgetting to divide the entire side by the coefficient, not just one term.
- Misreading the signs: subtracting b from d is not the same as subtracting d from b.
Equation Solver — frequently asked questions
What is a linear equation?
An equation where the variable appears only to the first power, like 2x + 4 = 1x + 7. Its graph is a straight line, hence the name.
How do I solve ax + b = cx + d?
Subtract cx from both sides, subtract b from both sides, then divide by the remaining coefficient of x: x = (d − b) ÷ (a − c).
What does 'no solution' mean?
The two sides describe parallel lines that never cross, so no value of x satisfies the equation. This happens when a = c but b ≠ d.
What does 'infinitely many solutions' mean?
The two sides are identical expressions, so every real number satisfies the equation. This happens when a = c and b = d.
Why do I need to check the answer?
Substitution confirms you made no arithmetic errors: plug x back in and both sides must produce the same number.
Can this solve equations with x on only one side?
Yes — just set c = 0, or d = 0 for the form ax + b = 0. The same steps isolate x automatically.