Linear Equation Solver (ax + b = c)

Last updated: 2026-05-19

What is a linear equation?

A linear equation in one variable is any equation that can be written as:

$$ax + b = c$$

where $a$, $b$, and $c$ are known numbers and $x$ is the unknown you want to find. "Linear" refers to the fact that $x$ appears to the first power — there are no $x^2$, $x^3$, or square-root terms. This is the simplest class of equation in algebra and the foundation of nearly every more advanced topic.

The formula: isolating x

To solve for $x$, apply the same operation to both sides of the equation until $x$ stands alone.

Step 1 — subtract $b$ from both sides:

$$ax = c - b$$

Step 2 — divide both sides by $a$ (provided $a \neq 0$):

$$x = \frac{c - b}{a}$$

This is the complete formula: substituting the three coefficients gives the value of $x$ directly.

Worked example

Solve $5x + 3 = 23$:

1. Identify coefficients: $a = 5$, $b = 3$, $c = 23$.
2. Subtract $b$: $5x = 23 - 3 = 20$.
3. Divide by $a$: $x = 20 / 5 = 4$.
4. Check: $5 \times 4 + 3 = 20 + 3 = 23$ ✓

Sign rules

When $b$ is negative — say, $b = -7$ — subtracting it becomes adding:

$$ax = c - (-7) = c + 7$$

And when $a$ is negative, dividing by a negative number flips the inequality sign (relevant in inequality solving, though not here). The solution $x = (c - b) / a$ handles both cases correctly regardless of sign.

The degenerate case when a = 0

When $a = 0$, the $x$ term vanishes and the equation reduces to $b = c$:

$b = c$ — infinitely many solutions. The equation says $0 + b = c$, which is simply $b = c$. If that's true, every real number $x$ satisfies the equation, because $x$ does not appear at all. Example: $0 \cdot x + 4 = 4$ is satisfied by $x = 1$, $x = 100$, $x = -\pi$, and every other real number.

$b \neq c$ — no solution. The equation asserts a false arithmetic fact — something like $4 = 7$. No value of $x$ can make a false statement true. Example: $0 \cdot x + 4 = 7$ has no solution.

Where linear equations appear

Linear equations show up constantly in everyday math and science:

• Distance / speed problems: $d = rt$ rearranges to $t = d / r$, a linear solve.
• Unit conversion: converting Celsius $C$ to Fahrenheit $F$ with $F = 1.8C + 32$ inverts to $C = (F - 32) / 1.8$.
• Break-even analysis: total revenue equals total cost at $r x = f + v x$, which rearranges linearly for $x$.
• Physics: Ohm's law $V = IR$, solving for current $I = V / R$.

In each case, isolating the unknown by undoing addition and multiplication — exactly the two steps above — is all that is needed.

Frequently Asked Questions (FAQ)

How do I solve ax + b = c?

Subtract b from both sides to get ax = c − b, then divide both sides by a to get x = (c − b) / a. This works as long as a ≠ 0. For example, solving 3x + 4 = 13 gives x = (13 − 4) / 3 = 9 / 3 = 3.

What happens if a is zero?

When a = 0 the equation reduces to b = c, which has nothing to do with x. If b and c are equal (e.g. 0·x + 5 = 5), every real number is a solution — infinitely many solutions. If b ≠ c (e.g. 0·x + 5 = 7), the equation is a false statement and there is no solution at all.

Can the solution be negative or fractional?

Yes. For instance, 2x + 7 = 3 gives x = (3 − 7) / 2 = −4 / 2 = −2 (negative), and 3x + 1 = 3 gives x = 2/3 (fractional). The calculator shows fractions automatically when the result is rational with a denominator up to 1000.

How is a linear equation different from a quadratic?

A linear equation (degree 1) has at most one unknown term — ax + b = c — and always has exactly one solution when a ≠ 0. A quadratic equation (degree 2) has an x² term, giving up to two roots. The linear case is the simplest building block: every quadratic or higher-degree polynomial reduces to a linear step during the final solution stage.