How to Solve Linear Equations Step by Step: A Guide to One- and Multi-Step Problems

Overview

A linear equation is an equation in which the variable has a degree of 1. Common examples include x + 7 = 15, 3x - 4 = 20, and 2(x + 5) = 18. The goal is to isolate the variable on one side of the equals sign while keeping the equation balanced.

The central rule is simple: whatever operation you perform on one side, perform on the other side as well. You may add, subtract, multiply, or divide both sides by the same number, provided you do not divide by zero. These operations are called inverse operations because they undo one another:

• Addition and subtraction are inverses.
• Multiplication and division are inverses.

For a broader reference of algebra rules and notation, see the math formula and equation reference sheet. If you are unsure which solving method fits a problem, the equation solver methods guide can help you choose an approach.

How to estimate

Before writing a solution, estimate what the variable should be. Estimation is not a replacement for algebra; it is a quick reasonableness check. For example, in 4x + 3 = 19, the value of x should be close to 4 because 4 times 4, plus 3, equals 19.

One-step equations

Use the inverse operation to isolate the variable.

Example: Solve x - 9 = 14.

1. Add 9 to both sides: x - 9 + 9 = 14 + 9.
2. Simplify: x = 23.

For a multiplication example, solve 6x = 42. Divide both sides by 6 to get x = 7.

Multi-step equations

For a typical multi-step equation, simplify each side first, move variable terms to one side if needed, move constants to the other side, and then divide by the coefficient of the variable.

Example: Solve 5x + 8 = 28.

1. Subtract 8 from both sides: 5x = 20.
2. Divide both sides by 5: x = 4.

The order matters less than the principle of keeping both sides balanced, but simplifying carefully at each line makes errors easier to spot.

Equations with parentheses

Use the distributive property before isolating the variable. Multiply the term outside the parentheses by every term inside.

Example: Solve 3(x + 4) = 27.

1. Distribute 3: 3x + 12 = 27.
2. Subtract 12: 3x = 15.
3. Divide by 3: x = 5.

A common mistake is distributing to only the first term inside the parentheses. Writing the multiplication explicitly can prevent that error.

Variables on both sides

When the variable appears on both sides, collect the variable terms on one side. Choose the side that keeps the coefficient positive when possible.

Example: Solve 7x - 2 = 3x + 18.

1. Subtract 3x from both sides: 4x - 2 = 18.
2. Add 2 to both sides: 4x = 20.
3. Divide by 4: x = 5.

If the variable terms cancel and the remaining statement is true, the equation has infinitely many solutions. If they cancel and the statement is false, the equation has no solution.

Inputs and assumptions

When solving a linear equation, identify the information that affects each step:

• Variable: The unknown quantity, such as x or y.
• Coefficients: Numbers multiplied by variables, such as the 4 in 4x.
• Constants: Numbers without variables.
• Operations: Addition, subtraction, multiplication, division, distribution, and combining like terms.
• Restrictions: Any value that would make a denominator zero must be excluded.

Fractions and decimals often look more difficult than they are because they add another representation of the same numbers.

Fractions

To remove fractions, multiply every term on both sides by a common denominator.

Example: Solve x/3 + 2 = 6.

1. Subtract 2: x/3 = 4.
2. Multiply both sides by 3: x = 12.

For several fractions, identify the least common denominator when practical, then multiply every term—not just the fraction you notice first.

Decimals

For terminating decimals, multiplying by a power of 10 can make the equation easier to read.

Example: Solve 0.4x + 1.2 = 3.6.

1. Multiply every term by 10: 4x + 12 = 36.
2. Subtract 12: 4x = 24.
3. Divide by 4: x = 6.

Alternatively, you can solve directly by subtracting 1.2 and dividing by 0.4. Either method is valid if the same operation is applied consistently.

Worked examples

Combining like terms

Solve 2x + 7 + 3x - 4 = 28.

1. Combine variable terms: 5x.
2. Combine constants: 3.
3. Rewrite the equation: 5x + 3 = 28.
4. Subtract 3: 5x = 25.
5. Divide by 5: x = 5.

Like terms must have the same variable part. For example, 2x and 3x are like terms, but x and x² are not.

Variables on both sides with distribution

Solve 2(x + 3) = x + 11.

1. Distribute: 2x + 6 = x + 11.
2. Subtract x from both sides: x + 6 = 11.
3. Subtract 6: x = 5.

Verification

Always substitute your answer into the original equation, not only the last simplified line. For 2(x + 3) = x + 11 and x = 5, the left side is 2(5 + 3) = 16, while the right side is 5 + 11 = 16. Since both sides match, the solution is verified.

Verification is especially useful when negative signs, fractions, or distribution are involved. A calculator or linear equation solver can check arithmetic, but writing the algebraic steps shows how the answer was obtained. For calculator-supported practice, review the guide to using a graphing calculator for algebra.

When to recalculate

Revisit your work whenever an answer does not fit the original equation, a sign changes unexpectedly, or a decimal result seems unreasonable. Start by substituting the answer back into the original equation. Then inspect the most error-prone steps: distributing a negative, combining unlike terms, clearing fractions, and moving a term across the equals sign.

Recalculate if the equation itself changes—for example, when a teacher corrects a copied number, a word problem supplies a different value, or you discover that parentheses or a denominator were omitted. Do not round too early. Keep fractions or several decimal places until the final step when precision matters.

For independent practice, use this routine:

1. Copy the equation carefully.
2. Estimate the likely size and sign of the answer.
3. Simplify both sides.
4. Isolate the variable using inverse operations.
5. Write one justified step per line.
6. Substitute the result into the original equation.
7. Compare your method with a trusted step-by-step solution or equation solver only after attempting the problem.

Once this process becomes familiar, one-step and multi-step equations follow the same underlying idea: preserve equality while undoing operations in a controlled order.