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Linear Equation Solver

Solve a linear equation of the form ax + b = c for x.

Result

x =
5

Solving 3x + 7 = 22 for x.

About the Linear Equation Solver

Solving for x in ax + b = c is one of the most fundamental skills in algebra, and it's easy to make a sign error rewiring the equation by hand. This calculator isolates x instantly and shows you exactly which equation it solved.

How It Works

You enter the coefficient a, the constant b, and the right-hand side c. The calculator rearranges ax + b = c algebraically to solve for x, or flags the special cases where a is zero.

x = (c − b) ÷ a

Formula & Methodology

Isolating x from ax + b = c takes two algebraic steps: first subtract b from both sides to get ax = c − b, then divide both sides by a to get x = (c − b) ÷ a. That second step is only valid when a is not zero, since division by zero is undefined. When a is zero, the equation reduces to just b = c, a statement with no x in it at all, so either it's always true (when b happens to equal c, giving infinite solutions) or never true (when b and c differ, giving no solution).

Step-by-Step: Calculating It By Hand

  1. 1Start with the equation ax + b = c.
  2. 2Subtract b from both sides to isolate the ax term: ax = c − b.
  3. 3Divide both sides by a to solve for x, provided a is not zero.
  4. 4If a is zero, check whether b equals c to determine infinite solutions or no solution.

Examples

Standard solve

For 3x + 7 = 22: subtracting 7 gives 3x = 15, and dividing by 3 gives x = 5.

No solution case

For 0x + 5 = 8: since a is zero and 5 doesn't equal 8, there's no value of x that works.

Advantages

  • Instantly solves for x with no manual algebra required
  • Correctly flags the special no-solution and infinite-solution cases
  • Shows the exact equation being solved to avoid confusion
  • Handles negative and fractional coefficients without issue

Common Mistakes

  • Sign errors when moving the constant b to the other side of the equation
  • Dividing by a before subtracting b, which produces the wrong result
  • Not recognizing when a equals zero produces a special case rather than a normal numeric answer
  • Forgetting to check whether the final answer actually satisfies the original equation

Edge Cases to Watch For

  • If a is zero and b equals c, every value of x satisfies the equation (infinite solutions).
  • If a is zero and b does not equal c, no value of x can satisfy the equation.
  • A negative coefficient a still solves normally, just be careful with sign handling when dividing.
  • Non-integer solutions are common and entirely valid; x doesn't need to be a whole number.

Common Use Cases

  • Algebra homework and coursework on solving for a single variable
  • Quick verification of hand-solved linear equations
  • Word problems that reduce to a linear equation in one unknown
  • Any situation needing a fast, reliable solve for x
Written & fact-checked by the Calculateus TeamLast updated August 5, 2026How we verify our formulas

Frequently asked questions

What does it mean if there's no solution?

If the coefficient of x is 0 and the remaining constants don't match (e.g. 0x + 5 = 8), no value of x can ever satisfy the equation - it's a mathematical contradiction, not a calculator error.

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Conclusion

A linear equation in one variable is often the first algebraic skill students master, and getting comfortable with it makes everything built on top, systems of equations, inequalities, and beyond, much easier. Our System of Equations Solver extends this same idea to two variables and two equations at once.

References