Linear equations: step-by-step guide, methods, examples, and practice
Use this page as the main linear-equations hub on CureMath — AI Math Explainer: understand what makes an equation linear, compare the common solve patterns, and then choose the right next page or tool.
Start here if you want the short version before reading the full method.
- A linear equation has the variable to the first power only.
- The goal is to isolate the variable by undoing operations in reverse order.
- A strong final check is to substitute the answer back into the original equation and confirm both sides match.
What this topic means and what to look for first.
Linear equations are one of the most common algebra entry points because the solving logic stays consistent even when the surface details change.
This guide helps you spot the equation pattern first, then choose the cleanest route instead of treating every example as if it had the same level of difficulty.
One reliable route through the topic.
- 1Write the equation clearly and identify the operations acting on the variable.
- 2Undo addition or subtraction before multiplication or division unless the structure clearly suggests another simplification first.
- 3If the equation contains fractions, decimals, or brackets, simplify that clutter before trying to isolate the variable fully.
- 4Keep the equation balanced by doing the same operation to both sides.
- 5Check the final value in the original equation rather than trusting the last line automatically.
Choose the route that fits the quadratic.
Best when a single inverse operation isolates the variable immediately.
Best when you need to undo more than one operation in the correct order before the variable is alone.
Best handled by simplifying the clutter first, then returning to the usual isolate-the-variable routine.
See the method in action.
x + 7 = 12
- Subtract 7 from both sides.
- This gives x = 5.
- Check by substitution: 5 + 7 = 12, so the answer works.
2x + 5 = 17
- Subtract 5 from both sides to get 2x = 12.
- Divide both sides by 2.
- So x = 6.
x/3 + x/6 = 3
- Use 6 as the common denominator and multiply every term by 6.
- This gives 2x + x = 18.
- So 3x = 18 and x = 6.
3(x + 2) = 18
- Expand the bracket to get 3x + 6 = 18.
- Subtract 6 to get 3x = 12.
- Divide by 3, so x = 4.
Things that commonly send the method off track.
- Doing an operation to only one side of the equation.
- Undoing multiplication or division before clearing addition or subtraction.
- Dropping a negative sign while simplifying.
- Checking the answer in a rearranged working line instead of the original equation.
Use a short verification pass before moving on.
- Substitute the final value into the original equation and see whether both sides match.
- If the check fails, retrace the first line where you moved or simplified a term rather than only staring at the final answer.
- If the equation had fractions or brackets, make sure the simplification step preserved the full structure of the original equation.
Try a few variations before switching to a calculator or solver tool.
- x - 4 = 9
- 3x + 8 = 20
- x/2 + 5 = 11
- 4(x - 1) = 20
- 0.5x + 1.2 = 3.7
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Extra algebra revision resources
If you want more printed algebra practice after this page, these broader searches are a sensible next step.
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Algebra workbook and revision book search
Useful if you want more equation, factorising, and worked-example practice in one printed source.
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GCSE algebra practice resources search
A wider GCSE-style search if you want more mixed algebra questions beyond one online guide.
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Short answers worth checking.
It is an equation where the variable appears only to the first power, so the graph would form a straight line.
Use inverse operations to undo what is happening to the variable, while keeping both sides balanced.
Substitute the final value back into the original equation and check that both sides are equal.
Simplify the clutter first, often by clearing denominators or scaling away awkward decimals, and then solve the simplified equation.
Continue with the next closely related topic.
Use the public site structure first, then switch into the solver tool only if you need a direct test.
CureMath uses artificial intelligence to suggest how a maths problem could potentially be solved. AI can make mistakes.
Check important answers independently before relying on them.