Two-step linear equation solver

This two-step linear equation solver isolates x in equations shaped like ax plus b equals c and keeps every result exact. Paste an equation such as 6x + 5 = 20, including ordinary integers, terminating decimals, or fractions. The result shows a normalized equation, the operation used to remove the constant, the division that isolates x, a reduced rational answer, and a substitution check. Because the arithmetic uses exact fractions instead of floating-point approximations, repeating answers remain trustworthy: one third stays 1/3 rather than becoming a rounded decimal. The solver is deterministic, runs without a network connection, and reports malformed or non-unique equations as clear input errors.

How a two-step equation is solved

A two-step linear equation has one variable term and one constant on the left, with a constant on the right: ax + b = c. The goal is to undo the operations around x in reverse order. First remove b by applying its inverse operation to both sides.

If b is positive, subtract b; if b is negative, add its absolute value. This produces ax = c - b while preserving equality because the same change was made on both sides. Next divide both sides by the nonzero coefficient a.

The result is x = (c - b) / a. The solver displays these transformations as ordered steps rather than returning only a final number. It also substitutes the answer into the original left side.

That final check should simplify to c, confirming that sign handling, subtraction, and division were consistent. A coefficient of zero cannot produce a unique x, so it is rejected instead of encouraging an invalid division by zero. This narrow focus makes the tool useful for learning the standard isolation method and for checking work without hiding the algebra behind a general symbolic engine.

Entering equations and reading exact results

Enter one equation with exactly one x, a signed constant term, and one equals sign. Spaces and an optional multiplication symbol before x are accepted, so 4x - 7 = 9 and 4*x-7=9 mean the same thing. You may use integers, terminating decimals, or fractions in every numeric position.

An omitted coefficient means one: x + 2 = 5 is read as 1x + 2 = 5, while -x + 2 = 5 uses a coefficient of negative one. Fractions use a slash, and their denominators must not be zero. The response includes coefficients a, b, and c as normalized reduced fractions.

The steps array records the original normalized equation, the inverse operation applied to the constant, and division by the coefficient. The check field substitutes the solution back into ax + b and shows the resulting right-hand value. The browser path uses the same pure solver logic, preserving exact output across both interfaces.

Fractions, signs, and common mistakes

Exact rational arithmetic matters whenever division does not terminate neatly in decimal notation. In 3x + 2 = 3, subtracting 2 gives 3x = 1, so the exact solution is 1/3. Converting that value early to 0.333 or a long floating approximation makes the substitution check imperfect and can cause later calculations to drift.

This solver instead stores every parsed decimal or fraction as an integer numerator over an integer denominator, reduces intermediate results by their greatest common divisor, and emits a reduced fraction. Pay close attention to negative constants. In 5x - 7 = 18, the first isolation step adds 7 to both sides; subtracting negative 7 is addition.

Negative coefficients are also valid, and dividing by one changes no value while dividing by negative one reverses the sign. Avoid adding an extra x term on the right or using parentheses, powers, or inequalities: those belong to broader equation tools and are intentionally outside this focused grammar. When checking homework, compare each inverse operation before comparing only the answer.

A correct final fraction reached through incorrect cancellation may hide a conceptual mistake. For automated use, inspect the structured numerator and denominator instead of parsing the displayed step text.