The core idea: isolate x

A linear equation is basically a balance scale: whatever you do to the left side, you must do to the right side. The whole game is to get x alone. That’s why “linear equation solving” is one of the first big wins in algebra — it teaches the rules that keep an equation true.

This calculator supports two popular one-variable formats: (1) ax + b = c and (2) ax + b = cx + d. Both are linear because x is only to the first power. No squares, no roots, no division by x.

Start with ax + b = c. Your goal is to remove the constant term b, then divide by the coefficient a.

• Step 1: Subtract b from both sides: ax = c − b
• Step 2: Divide both sides by a: x = (c − b) / a

That’s it — two moves. If you remember nothing else, remember the pattern: “move the constant, then divide by the x-coefficient.”

When x appears on both sides, you first collect x-terms on one side and constants on the other. One clean approach is: subtract cx from both sides, then subtract b from both sides.

• Step 1: Subtract cx from both sides: (a − c)x + b = d
• Step 2: Subtract b from both sides: (a − c)x = d − b
• Step 3: Divide: x = (d − b) / (a − c)

The key is the combined coefficient (a − c). If it becomes zero, you’ve hit a special case (more on that next).

No solution vs infinite solutions

Linear equations are friendly because they usually have a single answer. But sometimes x disappears. That can lead to either no solution (a contradiction) or infinitely many solutions (an identity).

• If a = 0, the equation becomes b = c.
• If b = c, then it’s always true → infinitely many solutions.
• If b ≠ c, it can never be true → no solution.

• If a − c = 0, the x-terms cancel, leaving b = d.
• If b = d, both sides match for every x → infinite solutions.
• If b ≠ d, you get something like 0x = 7 → no solution.

This is why teachers emphasize “check your steps” and “don’t divide by zero.” The calculator detects these automatically and explains the result clearly.

Worked examples you can copy

Solve: 2x − 4 = 10

• Start: 2x − 4 = 10
• Add 4 to both sides: 2x = 14
• Divide by 2: x = 7
• Check: 2(7) − 4 = 14 − 4 = 10 ✅

Solve: 3x + 2 = x + 10

• Subtract x from both sides: 2x + 2 = 10
• Subtract 2: 2x = 8
• Divide by 2: x = 4
• Check: 3(4)+2 = 14 and 4+10 = 14 ✅

Solve: 2x + 3 = 2x + 8

• Subtract 2x from both sides: 3 = 8
• That’s false → no solution

Solve: 5x − 10 = 5x − 10

• Subtract 5x: −10 = −10
• Always true → infinitely many solutions

What this solver does under the hood

This tool doesn’t “guess” x. It applies the same algebra you’d do by hand: (1) move constants, (2) move x-terms, and (3) divide by the final x-coefficient. For clarity, it generates a step list you can screenshot or copy.

It also supports fraction-style input (3/5, -7/2), because many classroom problems use exact rational numbers. When the final answer is a clean fraction, the solver shows both the fraction form and a decimal approximation.

After you solve, verify by substitution: plug x back into the original equation and see if both sides match. If they don’t match, the most likely issue is a sign flip or dividing only one side.

Linear equations have graphs that are straight lines. That’s why these equations usually have one solution: two non-parallel lines intersect at one point. In the “no solution” case, the lines are parallel. In the “infinite solutions” case, the lines are the same line.

Frequently Asked Questions

• What is a linear equation?

  It’s an equation where the variable is only to the first power (x¹). Examples include 2x − 3 = 9 or 4x + 1 = 2x − 7. If you see x², √x, or 1/x, it’s not linear.

• Does this solve equations with parentheses?

  This solver focuses on coefficient form inputs for speed and accuracy. If your original problem has parentheses, expand it first (distribute) to convert it into one of the supported forms. Example: 2(x − 3) = 10 becomes 2x − 6 = 10.

• Why does it say “no solution”?

  That happens when x cancels out and you’re left with a contradiction like 3 = 8. Geometrically, it means the lines never intersect (parallel lines).

• Why does it say “infinitely many solutions”?

  That happens when x cancels out and you’re left with a true statement like −10 = −10. Geometrically, it means both sides represent the same line.

• Should I use fractions or decimals?

  If your problem started with fractions, keep fractions to avoid rounding error. If it started with decimals, decimals are fine — just round consistently (e.g., 3–4 decimal places).

• How can I check my answer quickly?

  Substitute your x back into the original equation and compute both sides. If both sides match, you’re correct. If not, re-check sign changes and distribution.

Want more algebra help? Try the Quadratic Equation Solver and System of Equations Solver in the related links below.

Make linear solving feel automatic

If you want linear equations to become “easy points,” practice a tiny set of patterns repeatedly. The goal isn’t to memorize random steps — it’s to build instincts: move constants, collect x, then divide.

• Do 2 problems of the form ax + b = c.
• Do 2 problems of the form ax + b = cx + d.
• Do 1 edge-case problem to recognize “no solution” vs “infinite.”

• Budget math: Solve for how many items you can buy at a fixed price.
• Fitness math: Solve for calories per day given weekly targets.
• Work math: Solve for break-even point in simple profit equations.
• Game math: Solve for XP needed per day to hit a level.

That’s why a “simple” linear solver is surprisingly shareable: it’s a universal tool for turning “I need x” into an answer.