📐 Linear Equation: Calculator

Solve equations of the form ax + b = cx + d for x, with full working shown.

Solve equations of the form ax + b = cx + d for x.

SOLUTION
x = 0
Enter coefficients to see the step-by-step solution.

How the Linear Equation Calculator Works

A linear equation with one variable can always be rearranged into the form ax + b = cx + d, where a, b, c, and d are known numbers and x is the unknown you're solving for. This calculator isolates x algebraically and shows every step, which is useful for checking homework or understanding the process.

ax + b = cx + d
(a − c)x = d − b
x = (d − b) ÷ (a − c)  (provided a ≠ c)

Special Cases: No Solution and Infinite Solutions

If a equals c (the x-coefficients are the same on both sides) but b doesn't equal d, the x terms cancel out and you're left with a false statement — meaning the equation has no solution. If a equals c AND b equals d, both sides are identical for every value of x, meaning the equation is true for all real numbers (infinitely many solutions).

Working With Simpler Equations

A simpler equation like 2x + 5 = 11 fits this same form with c = 0 and d = 11 — just leave the right-side coefficient (c) at 0 to solve single-sided linear equations.

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❓ Frequently Asked Questions

What is a linear equation with one variable?
It's an equation where the unknown (usually x) appears only to the first power (no x², square roots of x, etc.), and can always be rearranged into the form ax + b = cx + d.
How do I solve an equation like 2x + 5 = 11?
Set c = 0 and d = 11 in this calculator (since there's no x term on the right side): (2 − 0)x = 11 − 5, so 2x = 6, and x = 3.
What does it mean if the calculator says 'no solution'?
This happens when the x-coefficients are equal on both sides (a = c) but the constants aren't (b ≠ d) — the x terms cancel out, leaving a mathematically false statement, so no value of x can satisfy the equation.
What does 'infinitely many solutions' mean?
This occurs when both the coefficients and the constants match exactly on both sides (a = c and b = d) — the equation is really the same statement written twice, so every real number is a valid solution for x.
Can this calculator handle decimal or negative coefficients?
Yes, all four values (a, b, c, d) accept decimals and negative numbers, and the calculator solves and shows steps correctly for any valid combination.

🏆 About This Tool — Accuracy & Trust

🔒 Data Privacy: All calculations run entirely within your browser using JavaScript. Nothing you enter here is ever transmitted to our servers, stored, sold, or shared.

📐 Accuracy Note: This tool uses standard, widely published formulas and guidelines. Results are estimates for informational purposes; for financial, medical, or engineering decisions, consult a licensed professional.

📅 Last Updated: September 2026.