On this page
Linear equations look intimidating when they're stacked with terms on both sides, but every single one — no matter how long — reduces to the same short list of moves. Once you can recognize those moves, you can solve any equation of the form ax + b = c without guessing. This guide covers the method from the ground up: what makes an equation "linear," the four core moves, eight worked examples of increasing difficulty (including fractions and decimals), the two edge cases every student eventually hits, the most common mistakes, and how to check your own answer before you turn it in. If you'd rather test yourself as you read, the free algebra solver shows the same steps for any equation you type in.
What makes an equation "linear"
A linear equation is any equation where the variable (usually x) appears only to the first power — no x², no x under a square root, no x in a denominator. That's what keeps it "linear": if you graphed both sides, you'd get a straight line. Because the variable never gets squared or multiplied by itself, there's exactly one value of x that makes the equation true (with a couple of edge-case exceptions covered below). This is different from a quadratic equation, where the variable is raised to the second power and there can be zero, one, or two solutions. For a more formal treatment of why linear equations graph as straight lines, Khan Academy's algebra basics course is a solid free reference.
The four moves you'll always use
- Simplify each side. Distribute any parentheses and combine like terms on the left and right independently, before touching the equals sign.
- Move variable terms to one side. Add or subtract terms so that every x ends up on one side of the equation.
- Move constants to the other side. Add or subtract plain numbers so that only the x term remains on one side.
- Divide by the coefficient. Whatever number is multiplying x, divide both sides by it to leave x alone.
That's the entire method. Everything else is bookkeeping — keeping track of signs and making sure whatever you do to one side of the equation, you do to the other.
Why the order matters
It's tempting to treat these four moves as interchangeable, but doing them out of order is where most avoidable mistakes creep in. Dividing before you've finished combining like terms means you're only dividing part of the expression, which produces the wrong coefficient. Distributing after you've already tried to move terms means you end up moving a piece of the expression that hasn't been fully simplified yet. Sticking to simplify → collect variables → collect constants → divide, in that exact order every time, removes an entire category of errors before they happen.
Example 1: A simple one-step move
Solve x + 7 = 12.
There's already only one term with x, so subtract 7 from both sides: x = 12 - 7 = 5. Done — x = 5.
Example 2: Two-step equation
Solve 2x + 5 = 15.
First move the constant: subtract 5 from both sides to get 2x = 10. Then divide both sides by 2: x = 5. This is the classic textbook shape, and it's a good one to practice until the two moves feel automatic.
Example 3: Variables on both sides
Solve 5x - 3 = 3x + 9.
Here, x shows up on both sides, so the first job is to get them together. Subtract 3x from both sides: 2x - 3 = 9. Now it looks like Example 2 — add 3 to both sides to get 2x = 12, then divide by 2: x = 6.
Example 4: Parentheses and distribution
Solve 3(x + 2) = 2x - 4.
Before you can move anything, distribute the 3 across the parentheses: 3x + 6 = 2x - 4. Now subtract 2x from both sides: x + 6 = -4. Subtract 6: x = -10. Notice that skipping the distribution step is the single most common source of errors in this type of problem — always expand parentheses fully before you start moving terms.
Example 5: Equation with a fraction
Solve x/3 + 4 = 10.
Subtract 4 from both sides: x/3 = 6. To undo division by 3, multiply both sides by 3: x = 18. If a coefficient is a fraction instead, like (2/3)x = 8, multiply both sides by its reciprocal (3/2) rather than dividing — same operation, less messy arithmetic.
Example 6: Negative coefficient
Solve -4x + 9 = 1.
Subtract 9 from both sides: -4x = -8. Divide by -4 — dividing a negative by a negative gives a positive: x = 2. Dropping the negative sign midway through is one of the most common sign errors; see the mistakes section below.
Example 7: Decimal coefficients
Solve 0.5x - 1.5 = 2.
Add 1.5 to both sides: 0.5x = 3.5. Divide by 0.5 (or equivalently, multiply by 2): x = 7.
Two special cases you'll eventually hit
Occasionally, the variable cancels out entirely when you try to combine both sides. There are two possible outcomes:
- No solution: if you end up with something false, like 3 = 7, the equation has no solution — there's no value of x that could ever make it true.
- Infinite solutions: if you end up with something always true, like 4 = 4, every value of x satisfies the equation (this happens when both sides were really the same expression written differently).
Example 8: A longer equation worked in full
Solve 4(2x - 1) - 5 = 3(x + 4) + 2x.
Start by distributing both sides: the left becomes 8x - 4 - 5, which simplifies to 8x - 9. The right becomes 3x + 12 + 2x, which simplifies to 5x + 12. The equation is now 8x - 9 = 5x + 12 — the same shape as the earlier examples. Subtract 5x from both sides: 3x - 9 = 12. Add 9 to both sides: 3x = 21. Divide by 3: x = 7. Notice that the extra complexity here was entirely in the simplification stage — once both sides were fully expanded and combined, it was a two-step equation like any other.
Common mistakes (and how to fix them)
- Losing a negative sign when distributing. -2(x - 5) becomes -2x + 10, not -2x - 10. Distribute the sign along with the number.
- Adding instead of subtracting to move a term. If a term is +3x on the right and you want it gone, subtract 3x from both sides — don't just cross it out.
- Dividing only one term by the coefficient. In 2x + 4 = 10, after subtracting to get 2x = 6, divide the whole side by 2 — not just the x.
- Forgetting to apply an operation to both sides. Whatever you do to isolate x must happen to the entire other side, not just one term of it.
For a deeper breakdown of these and other slip-ups, see Common Algebra Mistakes Students Make (And How to Fix Them).
How to check your answer
Once you have a value for x, plug it back into the original equation — not a simplified version — and confirm both sides come out equal. For Example 4 above: 3(-10 + 2) = 3(-8) = -24, and 2(-10) - 4 = -20 - 4 = -24. Both sides match, so x = -10 is confirmed correct. SolveStep automatically checks every solved equation by substitution — but it's worth doing by hand too, since it catches exactly the sign and arithmetic errors described above before they cost you points.
Linear equations vs. other equation types
- Linear vs. quadratic: a linear equation has x to the first power and one solution; a quadratic has x² and up to two solutions. See the quadratic equations guide for the next step up.
- Linear equation vs. linear inequality: an inequality uses <, >, ≤, ≥ instead of =, and the four moves are nearly identical — with one extra rule: flip the inequality sign if you multiply or divide both sides by a negative number.
- One variable vs. systems of equations: everything above solves for a single unknown. Two equations and two unknowns (like x and y) need substitution or elimination instead — a related but different method.
The fastest way to build confidence with linear equations is to solve the same equation two ways: by hand, then check it against a tool that shows its steps, so you can see exactly where your working matches — or doesn't.
Practice with instant feedback
If you want to check your own working against a full step-by-step breakdown, type any linear equation into the SolveStep algebra solver — it walks through the same four moves described above and shows the running result after each one, then checks the answer by substitution automatically. Want unlimited fresh problems instead of typing your own? The practice mode generates random linear equations on demand.
→ Try the free step-by-step solver nowFAQs
What is a linear equation, in simple terms?
An equation where the variable appears only to the first power, so it graphs as a straight line and has exactly one solution in most cases.
What are the 4 steps to solving a linear equation?
Simplify both sides, move variable terms to one side, move constants to the other side, then divide by the coefficient of x.
Can a linear equation have no solution?
Yes — if simplifying both sides leaves a false statement (like 3 = 7), there's no value of x that works.
Can a linear equation have infinite solutions?
Yes — if simplifying both sides leaves a true statement (like 4 = 4), every value of x satisfies the equation, because both sides were the same expression in different forms.
Do I need to flip the sign when solving a linear equation?
Not for an equation with an equals sign. You only flip a comparison sign when solving an inequality and you multiply or divide both sides by a negative number.