Linear Equation Solver
Solve linear equations of the form ax + b = cx + d. Enter the four coefficients, the solver isolates x and shows each move.
How to use this calculator
Enter the coefficients a, b, c, d for your equation ax + b = cx + d and press Calculate. The solver moves x-terms left and constants right, combines them, divides once, and then checks the answer by substituting it back into both sides.
How it works
Solving a linear equation means isolating x: whatever you do to one side, do to the other. Collect the x-terms on the left (a − c)x and the constants on the right (d − b), then divide. Two special cases exist: if the x-terms cancel and the constants match, every x works (identity); if they cancel and the constants differ, no x works (contradiction). The solver detects both.
Worked example
Example: 2x + 3 = 11
Step 1, Move the constant: 2x = 11 − 3 = 8.
Step 2, Divide: x = 8 ÷ 2 = 4.
Step 3, Check: left side 2(4)+3 = 11, right side 11. Match ✓.
Common mistakes
- Sign errors when moving terms: +3 becomes −3 across the equals sign. Track every sign flip.
- Dividing only part of the side: in (2x + 6) = 10, dividing “just the 2x” by 2 is wrong, divide the whole side.
- Assuming x always exists: 2x + 3 = 2x + 7 has no solution; 2x + 3 = 2x + 3 has infinitely many. Both are valid answers.
When to use it
Linear equations are the workhorse of applied math: break-even points, unit pricing, mixture problems, distance-rate-time, and every “find the unknown quantity” word problem that does not involve squares or roots.
Frequently asked questions
Isolate x: move all x-terms to one side and constants to the other (flipping signs), combine like terms, then divide by the coefficient of x. Check by substituting back.
If simplifying leaves a true statement like 5 = 5, every x is a solution (identity). If it leaves a false one like 3 = 7, no x is a solution (contradiction).
Yes, enter coefficients as fractions (1/2) or decimals. The engine parses both and keeps the steps exact where possible.
That is exactly what the a, b, c, d form handles: ax + b = cx + d collects the x-terms first, then solves as usual.
Solve for x with steps
Our algebra solver shows every step as it isolates x: inverse operations applied to both sides, with the check substitution at the end.