On this page
A linear equation with terms stacked on both sides looks harder than it is. However long it runs, it reduces to the same short list of moves, and once you can recognise them you can solve anything 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 mistakes that cost the most marks, and how to check your own answer before you hand 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 one in which the variable, usually x, appears only to the first power. No x², no x under a square root, no x in a denominator. That is what makes it "linear": graph either side and you get a straight line. Because the variable is never squared or multiplied by itself, exactly one value of x makes the equation true, with two edge cases covered below. A quadratic equation behaves differently: 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 is the entire method. Everything else is bookkeeping: keeping track of signs, and making sure that whatever you do to one side of the equation you also 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. That is the answer: 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. Skipping the distribution step is the single most common source of errors in this type of problem, so 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. It is the same operation with tidier arithmetic.
Example 6: Negative coefficient
Solve -4x + 9 = 1.
Subtract 9 from both sides: -4x = -8. Divide by -4, and since a negative divided by a negative is 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, because no value of x 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. All the extra difficulty here sat in the simplification stage. Once both sides were fully expanded and combined, it became 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 rather than crossing 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 the x alone.
- 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, put it back into the original equation, not a simplified version of it, 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 checks every solved equation by substitution automatically, and it is still worth doing by hand, because that is what catches the sign and arithmetic errors described above before they cost you marks.
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, say x and y, need substitution or elimination instead, which is a related but separate 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 the tool's, and where it does not.
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, shows the running result after each one, then checks the answer by substitution. 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.