Linear Equation Solver

This linear equation solver finds x for any first-degree equation written as ax + b = cx + d. Type the four coefficients and the answer appears instantly, together with a clear step-by-step breakdown of how each term is moved, combined, and divided.

A linear equation has the variable x raised only to the first power. After the x terms move to one side and the constants to the other, there is at most one value of x that makes both sides equal.

Private by design: every calculation runs locally in your browser — nothing you type is uploaded or stored.

Who It Is For

Homework & Study

Check algebra answers and learn the method by following every intermediate step.

Quick Calculations

Isolate x in seconds for word problems, conversions, or everyday math.

Concept Practice

See special cases like infinite solutions and no solution to deepen your understanding.

How to Solve for X Step by Step

Enter the Coefficients

Type the values of a, b, c, d for ax + b = cx + d. Leave any field empty to treat that coefficient as 0, and use decimals such as 1.5 or -0.25 when needed.

Check the Live Preview

The equation preview updates in real time as you type, so you can confirm you entered ax + b = cx + d exactly the way you intended.

Read the Result

The value of x appears automatically — shown as an exact fraction when it is not a whole number — along with a clear status when the equation has infinite or no solutions.

Follow the Steps

Read the numbered breakdown to see exactly how terms are grouped, combined, and divided to isolate x.

The Formula

Moving every x term to the left and every constant to the right gives (a − c)x = d − b. Dividing both sides by (a − c) isolates the variable:

x = (d − b) / (a − c), valid whenever a ≠ c.

Worked Examples

Equation	Coefficients	Working	Result
2x + 3 = 7	a=2, b=3, c=0, d=7	(7 − 3) / (2 − 0) = 4 / 2	x = 2
3x − 1 = x + 5	a=3, b=−1, c=1, d=5	(5 − (−1)) / (3 − 1) = 6 / 2	x = 3
5x = 10	a=5, b=0, c=0, d=10	(10 − 0) / (5 − 0) = 10 / 5	x = 2
x + 1 = x + 1	a=1, b=1, c=1, d=1	(1 − 1) / (1 − 1) = 0 / 0	∞ solutions

Features

Instant Solving

Results appear the moment you change a coefficient — there is no submit button.

Live Equation Preview

The equation ax + b = cx + d rebuilds as you type so you can verify your input at a glance.

Step-by-Step Solutions

Every move — grouping terms, combining like terms, and dividing — is laid out as numbered steps.

Fraction Results

Answers appear as exact fractions, with the decimal value shown alongside, when x is not whole.

Special Case Detection

Automatically recognizes equations with infinite solutions or no solution at all.

Runs in Your Browser

All math is computed locally — no accounts, no uploads, and it works instantly.

Understanding the Three Outcomes

Outcome	Condition	Meaning
Unique solution	`a ≠ c`	Exactly one value of `x` satisfies the equation.
Infinite solutions	`a = c` and `b = d`	Both sides are identical (e.g. `0 = 0`); every `x` works.
No solution	`a = c` and `b ≠ d`	A contradiction appears (e.g. `0 = 5`); no `x` works.

Frequently Asked Questions

What is a linear equation?

A linear equation is one where the highest power of the variable is 1, such as 2x + 3 = 7. Its graph is a straight line, which is where the name "linear" comes from.

How do I solve an equation with variables on both sides?

Enter it as ax + b = cx + d, putting the left-side coefficients in a, b and the right-side ones in c, d. The solver moves all x terms to one side and constants to the other, giving (a − c)x = d − b, then divides to find x.

When does a linear equation have no solution?

When simplifying leads to a contradiction like 0 = 5. This happens when the x terms cancel (a = c) but the constants differ (b ≠ d), so no value of x can satisfy the equation.

When does it have infinite solutions?

When both sides simplify to the same expression (e.g. 0 = 0). The x terms cancel (a = c) and the constants are equal (b = d), so every value of x is a valid solution.

Can I leave some coefficients blank?

Yes. Any empty field is treated as 0. For example, to solve 5x = 10 you only need a = 5 and d = 10, leaving b and c empty.

Does it support decimal coefficients and fraction answers?

Yes. You can enter decimal values such as 1.5 or -0.25, and the solver returns the answer as an exact fraction — with the decimal shown alongside — whenever the result is not a whole number.