How to Solve Linear Equations: Step-by-Step with Examples

Linear equations are the foundation of algebra. Every other algebra topic, systems, inequalities, functions, even quadratics, rests on the ability to solve a linear equation cleanly. The good news: the process is mechanical. If you know the steps, you can solve any linear equation in under a minute.

This guide takes you from “\(x + 3 = 7\)” all the way through “\(3(x – 4) = 5x + 2 – x\),” with every worked step.

What a Linear Equation Is

A linear equation is an equation where the variable appears to the first power, no exponents, no roots, no \(x^2\). Examples:

• \(x + 3 = 7\)
• \(3x – 5 = 16\)
• \(2(x + 1) = x – 3\)
• \(\dfrac{x}{4} + 2 = 9\)

The goal is always the same: isolate the variable on one side of the equal sign.

The One Big Rule

Whatever you do to one side, do to the other. The equal sign is a balance scale. Add 3 to the left, you add 3 to the right.

The operations you can use: 1. Add the same number to both sides. 2. Subtract the same number from both sides. 3. Multiply both sides by the same nonzero number. 4. Divide both sides by the same nonzero number.

That’s it. Every linear equation reduces to those four operations.

One-Step Equations

Pattern: One operation between the variable and a number.

Example 1

\(x + 7 = 12\) Subtract 7 from both sides: \(x = 5\).

Example 2

\(x – 3 = 10\) Add 3 to both sides: \(x = 13\).

Example 3

\(5x = 30\) Divide both sides by 5: \(x = 6\).

Example 4

\(\dfrac{x}{4} = 8\) Multiply both sides by 4: \(x = 32\).

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Two-Step Equations

Pattern: Two operations between the variable and the answer.

The key: undo addition or subtraction first, then undo multiplication or division. (This is the reverse order of operations.)

Example 5

\(3x + 5 = 14\) Step 1: subtract 5: \(3x = 9\). Step 2: divide by 3: \(x = 3\).

Example 6

\(\dfrac{x}{2} – 4 = 7\) Step 1: add 4: \(\dfrac{x}{2} = 11\). Step 2: multiply by 2: \(x = 22\).

Example 7

\(-2x + 5 = 13\) Step 1: subtract 5: \(-2x = 8\). Step 2: divide by $-2$: \(x = -4\).

Always check by plugging back: \(-2(-4) + 5 = 8 + 5 = 13\). ✓

Multi-Step Equations

Pattern: More than two operations, often with parentheses or like terms.

Steps for any multi-step equation

1. Distribute any multiplication into parentheses.
2. Combine like terms on each side.
3. Move variable terms to one side.
4. Move constant terms to the other side.
5. Divide to isolate the variable.

Example 8

\(2(x + 3) = 10\) Distribute: \(2x + 6 = 10\). Subtract 6: \(2x = 4\). Divide by 2: \(x = 2\).

Example 9

\(3(x – 2) + 4 = 19\) Distribute: \(3x – 6 + 4 = 19\). Combine: \(3x – 2 = 19\). Add 2: \(3x = 21\). Divide: \(x = 7\).

Example 10

\(5x + 3 = 2x + 18\) Subtract $2x$: \(3x + 3 = 18\). Subtract 3: \(3x = 15\). Divide: \(x = 5\).

Example 11 (with parentheses on both sides)

\(2(x + 3) = 4(x – 1)\) Distribute: \(2x + 6 = 4x – 4\). Subtract $2x$: \(6 = 2x – 4\). Add 4: \(10 = 2x\). Divide: \(x = 5\).

Equations with Fractions

The trick: multiply both sides by the LCD (least common denominator) to clear fractions first.

Example 12

\(\dfrac{x}{3} + \dfrac{1}{2} = 2\) LCD is 6. Multiply both sides by 6: \(2x + 3 = 12\). Subtract 3: \(2x = 9\). Divide: \(x = 4.5\).

Example 13

\(\dfrac{x + 2}{4} = \dfrac{x – 1}{3}\) Cross-multiply: \(3(x + 2) = 4(x – 1)\). Distribute: \(3x + 6 = 4x – 4\). Subtract $3x$: \(6 = x – 4\). Add 4: \(x = 10\).

Special Cases

No solution

\(2x + 3 = 2x + 5\) → subtract $2x$: \(3 = 5\). False statement → no solution.

Infinite solutions

\(3(x + 2) = 3x + 6\) → distribute: \(3x + 6 = 3x + 6\). Always true → all real numbers.

Both cases mean the original equation is not really a linear equation in one variable. It’s an identity (always true) or a contradiction (never true).

Common Mistakes

Forgetting the negative sign

\(-3(x – 4) = -3x + 12\), not $-3x – 12$. Distribute carefully.

Combining unlike terms

$3x + 5$ is not $8x$. You can only combine $3x$ with another \(x\) term.

Dropping the equal sign

Every step should re-write the full equation. Don’t lose track of what equals what.

Forgetting to divide both sides

\(2x = 14\) → \(x = 7\), not \(x = 14\).

Not checking the answer

Plug your answer back in. If both sides are equal, you’re done. If not, find the mistake.

Mishandling decimals

\(0.5x = 3\) → \(x = 6\), not \(x = 1.5\). Divide both sides by 0.5.

How to Practice

1. Start with one-step equations. Build automatic fluency.
2. Move to two-step. Drill until each takes under 30 seconds.
3. Mix in distribution and fractions. Slow down, write every step.
4. Always check your answer. Always.
5. Do 5 problems a day for 2 weeks. That’s 70 problems total, enough to lock in mastery.

Free Resources

Effortless Math has a complete free linear-equation library:

• Algebra 1 Worksheets, equation problems by difficulty with answer keys.
• Math Topics Library, every linear-equation topic explained.
• Algebra 1 eBooks, full Algebra 1 workbooks.

Frequently Asked Questions

How do you solve linear equations step by step? First simplify each side by distributing and combining like terms. Next, move the variable terms to one side and the constants to the other, then divide by the coefficient. For 3(x − 4) = 5x + 2 − x, distribute to get 3x − 12 = 4x + 2. Subtract 3x to get −12 = x + 2, then subtract 2: x = −14. Check: 3(−18) = −54 and 4(−14) + 2 = −54.

How do you solve linear equations with fractions? Clear the fractions first by multiplying every term on both sides by the least common denominator. For x/3 + 1/2 = 5/6, the LCD is 6, so the equation becomes 2x + 3 = 5. Subtract 3 to get 2x = 2, then divide by 2: x = 1. Be sure to multiply every term, including whole numbers, or the equation will no longer balance.

How do you solve linear equations with variables on both sides? Gather the variable terms on one side by adding or subtracting, then solve the two-step equation that remains. For 7x − 4 = 3x + 12, subtract 3x from both sides to get 4x − 4 = 12. Add 4 to get 4x = 16, then divide by 4: x = 4. Moving the smaller x term usually keeps the coefficient positive and helps you avoid sign errors.

What does no solution mean in linear equations? A linear equation has no solution when the variable terms cancel and leave a false statement. For 2x + 5 = 2x − 1, subtracting 2x from both sides gives 5 = −1, which is never true, so no value of x works. If the variables cancel and leave a true statement instead, such as 4 = 4, the equation has infinitely many solutions.

What is solving linear equations in one variable? It means finding the single value of the variable that makes both sides of the equation equal. In a linear equation the variable appears only to the first power, as in 3x − 5 = 16. Use inverse operations and do the same thing to both sides: add 5 to get 3x = 21, then divide by 3 to get x = 7.

Is solving linear equations hard? Solving linear equations gets much easier once you learn the routine, because it uses only four moves: add, subtract, multiply or divide both sides by the same number. Most mistakes come from sign errors, forgetting to distribute to every term, or changing only one side. Build up from one-step to two-step to multi-step equations, and check every answer by substituting it back in.

You’ve Got the Method

Linear equations are not memorization. They are a system. Learn the four operations, follow the order, check your answer. Every algebra and pre-calc course rests on this skill. Master it now and the rest of math gets easier.

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