Solve a linear equation in one variable, with step-by-step working. Two modes: the simple form ax + b = c where x only appears on one side, and the two-sided form ax + b = cx + d where x appears on both. The solver handles all three possible outcomes — a unique solution, no solution (contradiction), or infinite solutions (identity) — and shows each transformation as a numbered step so you can check the work or copy it into a homework answer.
Simple mode: ax + b = c
This is the classic isolation problem. Given a, b, c, find x. The algorithm is: subtract b from both sides, then divide both sides by a.
- Unique solution when a ≠ 0. You get x = (c − b) / a.
- No solution when a = 0 and b ≠ c. The equation reduces to “0 = c − b” which can’t be true.
- Infinite solutions when a = 0 and b = c. The equation reduces to “0 = 0” which is always true.
The middle two cases are edge cases of the formula — they happen when the coefficient of x itself is zero, so “dividing by a” would be division by zero. The tool handles them explicitly and surfaces the result as a text label rather than returning NaN.
Two-sided mode: ax + b = cx + d
This is the more general form. The algorithm is: subtract cx from both sides, then subtract b, then divide.
- Unique solution when a ≠ c. You get x = (d − b) / (a − c).
- No solution when a = c and b ≠ d. The lines are parallel (same slope, different y-intercept).
- Infinite solutions when a = c and b = d. The two sides describe the same line.
The two-sided case is just a rearrangement of the simple case: rewrite ax + b = cx + d as (a − c)x = d − b, then you have a simple equation with coefficient (a − c) and right-hand side (d − b). The tool does the rearrangement for you and shows the resulting intermediate form as part of the step list.
Example: isolating a variable
2x + 3 = 11. Simple mode, a = 2, b = 3, c = 11. Steps:
- Start: 2x + 3 = 11
- Subtract 3 from both sides → 2x = 8
- Divide both sides by 2 → x = 4
Check: 2(4) + 3 = 11. ✓
Example: a two-sided problem
3x + 2 = x + 10. Two-sided mode, a = 3, b = 2, c = 1, d = 10. Steps:
- Start: 3x + 2 = 1x + 10
- Subtract x from both sides → 2x + 2 = 10
- Subtract 2 from both sides → 2x = 8
- Divide both sides by 2 → x = 4
Check: 3(4) + 2 = 14, 1(4) + 10 = 14. ✓
Example: no solution (parallel lines)
2x + 3 = 2x + 5. Two-sided mode, a = 2, b = 3, c = 2, d = 5. Steps:
- Start: 2x + 3 = 2x + 5
- Subtract 2x from both sides → 0x + 3 = 5
- Subtract 3 from both sides → 0 = 2
- Contradiction — no value of x works
Graphically: the two lines y = 2x + 3 and y = 2x + 5 have the same slope but different y-intercepts, so they’re parallel and never meet. A word problem that reduces to this pattern usually means two constraints are incompatible.
Example: infinite solutions (identity)
2x + 6 = 2(x + 3). Expand the right side first: 2x + 6 = 2x + 6. Two-sided mode, a = 2, b = 6, c = 2, d = 6. Steps:
- Start: 2x + 6 = 2x + 6
- Subtract 2x → 0x + 6 = 6
- Subtract 6 → 0 = 0
- Identity — any value of x works
This comes up when one equation is just a restatement of another, or when you simplify a problem to the point where the variable is no longer constrained.
How the step-by-step is generated
The tool runs the solver, but it doesn’t just compute the answer — it records each transformation as a human-readable line, in the same order a maths teacher would write them on a whiteboard. Subtracting b. Dividing by a. Reaching a contradiction or identity. Each line is a plain numeric expression, no LaTeX, no prose — it’s meant to drop straight into homework or be copied to paper.
What this tool does not do
It doesn’t parse equations as strings. You can’t paste “2x + 3 = 11” into a single input and have it figure out the coefficients. You have to decompose the equation yourself and type the numeric coefficients. For string parsing, a computer algebra system like WolframAlpha or SymPy is the right tool.
It doesn’t solve quadratic, cubic, or higher-degree equations. This is strictly first-degree (linear) in one variable. For quadratics, use the quadratic equation solver. For higher degrees, use a numerical root finder.
It doesn’t solve systems of equations — two equations in two unknowns, three equations in three unknowns, and so on. That’s a separate problem, handled by the system of equations solver in the same category. This tool is for single equations in a single variable.
It doesn’t show graphs. The step-by-step shows the algebra, but not the geometric interpretation. If you want to see the lines meeting at a point (or not meeting, or overlapping), use a graphing tool.
It doesn’t handle symbolic coefficients. Every input must be a plain number. You can’t type “pi” or “sqrt(2)” — compute the numeric value first, then enter it.