Learning how to solve linear equations step by step gives you a reliable method for finding an unknown value instead of relying on guesswork. This guide covers one-step, two-step, and multi-step equations, including equations with parentheses, fractions, decimals, and variables on both sides. Use the checklists and worked examples as a reusable reference for homework, practice, and test preparation.
Overview
A linear equation is an equation in which the variable has a first power, such as 3x + 5 = 20. The goal is to isolate the variable on one side of the equal sign. The answer is the value that makes the original equation true.
The central principle is balance: whatever operation you perform on one side of an equation, perform on the other side as well. Inverse operations help undo the operations attached to the variable:
- Addition and subtraction undo each other.
- Multiplication and division undo each other.
A dependable general order is:
- Simplify each side by distributing and combining like terms.
- Move variable terms to one side when necessary.
- Move constant terms to the other side.
- Divide or multiply to isolate the variable.
- Substitute the result into the original equation to check it.
This process works for most ordinary linear equations. If an equation contains two variables, it may be a system rather than a single linear equation. For those problems, see the systems of equations solver methods guide.
Checklist by scenario
One-step equations
Use the inverse operation once to isolate the variable.
Example: Solve x + 9 = 16.
- Subtract 9 from both sides: x + 9 - 9 = 16 - 9.
- Simplify: x = 7.
- Check: 7 + 9 = 16, which is true.
For an equation such as 5x = 35, divide both sides by 5. The result is x = 7.
Two-step equations
Undo addition or subtraction first, then undo multiplication or division.
Example: Solve 4x - 6 = 18.
- Add 6 to both sides: 4x = 24.
- Divide both sides by 4: x = 6.
- Check: 4(6) - 6 = 24 - 6 = 18.
Writing one operation per line makes the work easier to inspect and reduces sign errors.
Multi-step equations with parentheses
Start by using the distributive property. Multiply every term inside the parentheses by the number outside.
Example: Solve 3(2x - 4) + 5 = 23.
- Distribute 3: 6x - 12 + 5 = 23.
- Combine like terms: 6x - 7 = 23.
- Add 7: 6x = 30.
- Divide by 6: x = 5.
- Check in the original equation: 3(10 - 4) + 5 = 18 + 5 = 23.
Be especially careful when the factor outside parentheses is negative. For example, -2(x + 3) becomes -2x - 6, not -2x + 6.
Variables on both sides
Collect all variable terms on one side before isolating the variable.
Example: Solve 7x - 4 = 3x + 20.
- Subtract 3x from both sides: 4x - 4 = 20.
- Add 4 to both sides: 4x = 24.
- Divide by 4: x = 6.
You could instead move the 7x term to the right. Either choice is valid; choose the direction that keeps coefficients positive or makes the arithmetic simpler.
Equations with fractions
When several terms contain fractions, multiply every term by the least common denominator. This clears the fractions before you continue.
Example: Solve x/3 + 2 = 6.
- Subtract 2: x/3 = 4.
- Multiply both sides by 3: x = 12.
- Check: 12/3 + 2 = 4 + 2 = 6.
For a more involved equation such as x/4 - (x + 2)/3 = 5, multiply every term by 12, not just the terms that visibly contain a fraction.
Equations with decimals
To remove decimals, multiply every term by a power of 10. Choose 10, 100, or another power based on the greatest number of decimal places.
For 0.4x + 1.2 = 3.6, multiply the entire equation by 10:
4x + 12 = 36, so 4x = 24 and x = 6.
Keeping the decimal points aligned can also help when adding or subtracting decimal terms.
What to double-check
A correct-looking procedure can still produce a wrong answer if a sign or factor is mishandled. Before submitting a solution, use this short review:
- Distribution: Did the outside factor multiply every term inside the parentheses?
- Signs: When you moved a term across the equal sign, did its sign change correctly?
- Like terms: Did you combine only terms with the same variable part?
- Fractions: If you multiplied by a denominator, did you multiply every term on both sides?
- Division: Did you divide the entire side by the coefficient, not just part of it?
- Original equation: Does the answer make the starting equation true?
For example, if your work gives x = -2, substitute -2 into the original equation rather than a simplified intermediate line. This catches errors that may have been carried forward. A calculator or equation calculator can verify arithmetic, but it should support the written reasoning rather than replace it when the assignment requires showing work.
Common mistakes
Changing a sign without applying the operation to both sides
From x + 8 = 13, you cannot simply erase the 8. Subtract 8 from both sides to preserve equality.
Distributing to only the first term
The expression 5(x - 2) equals 5x - 10. Forgetting to multiply -2 changes the equation.
Combining unlike terms
Terms such as 3x and 7 are not like terms. They cannot be combined into 10x. The variable and constant parts must match appropriately.
Dividing only one term
In 6x + 12 = 30, first subtract 12. Then divide the entire equation 6x = 18 by 6. Dividing 30 but not 18 would break the balance.
Stopping without a check
A check takes only a few seconds and is particularly valuable when negatives, fractions, or decimals are involved. Keep a record of recurring errors in an equation practice planner so your next study session targets the actual difficulty.
When to revisit
Return to this checklist whenever a new equation format appears, such as variables on both sides, nested parentheses, fractions, or decimals. Revisit it before a quiz or test after completing a few practice problems without looking at the examples. Your goal is not merely to remember a sequence; it is to recognize which operation should happen next.
For a practical review routine, solve one one-step equation, one two-step equation, one distribution problem, one variable-on-both-sides problem, and one fraction or decimal equation. Check each answer in the original equation and note the first step where an error occurs. If you need broader practice, combine this guide with a math homework planner or the linear equations calculator guide for additional checking strategies.
Finally, keep the core checklist visible: simplify, collect terms, isolate the variable, and check the answer. With consistent written steps, solving for x becomes a repeatable process rather than a collection of unrelated tricks.
