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Updated Mar 19, 2026

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12 pages

Step-by-Step Guide to Solving Linear Equations in One Variable

Linear equations are everywhere - from calculating how much change... Show more

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Solving linear equations in one variable - step by step guide
Master techniques for solving linear equations with one
variable
What you will

What Makes an Equation Linear?

You've probably solved equations before without realising they were linear equations! A linear equation has just one variable (like x or y) and that variable is never squared or cubed - it's always to the power of 1.

Think of these equations like a straight line on a graph (that's why they're called 'linear'). The basic form looks like this: ax + b = c, where a, b, and c are just numbers.

Here are some examples you might recognise: 2x + 5 = 11 or 3y - 7 = 14. Notice how the variable appears only once and isn't squared or in fractions.

Key Point: If you can write your equation as ax + b = c, you've got a linear equation - and that means you can solve it!

The Balance Method

Imagine an equation as a perfectly balanced scale - whatever you do to one side, you must do to the other side to keep it balanced. This is your golden rule for solving any linear equation.

Inverse operations are your best friends here. Addition undoes subtraction, and multiplication undoes division. If you see +5 on one side, subtract 5 from both sides. If you see ÷3, multiply both sides by 3.

Let's try 2x + 7 = 15: First, subtract 7 from both sides to get 2x = 8. Then divide both sides by 2 to get x = 4. Always check your answer by putting it back into the original equation!

Top Tip: The balance method never fails - just remember to do the same thing to both sides every single time.

Tackling Tricky Equations

When variables appear on both sides of an equation, don't panic! Just collect all the x terms on one side and all the numbers on the other side.

For equations with brackets, use the distributive property first. If you see 2 $x + 3$ , multiply both terms inside: 2x + 6. Then solve normally.

Fractions in equations might look scary, but there's a simple trick. Multiply everything by the bottom number of the fraction to clear it away. For x/3 + 2 = 5, multiply everything by 3 to get x + 6 = 15.

Remember: Take it step by step - simplify first, then move terms around, and finally isolate your variable.

Real-World Problem Solving

Linear equations help you solve actual problems you might face! Whether you're working out costs at a shop or calculating distances, the same problem-solving steps apply.

Start by reading carefully and deciding what your variable represents. If Maria buys notebooks for £15 each plus a £25 pen, and spends £85 total, then 15x + 25 = 85 tells you she bought 4 notebooks.

Transport problems work the same way. If a taxi charges £12 for the first 4km and £2 for each extra km, you can work out any journey cost using linear equations.

Success Strategy: Define your variable clearly, write your equation based on the story, solve it step by step, and always check your answer makes sense!

Avoiding Common Mistakes

The biggest mistake students make is forgetting the balance rule - always do the same operation to both sides! If you subtract 5 from the left side, you must subtract 5 from the right side too.

Sign errors trip up loads of students. When you move +7 from one side to the other, it becomes -7. Think of it as the term changing its sign as it crosses the equals sign.

When distributing, multiply the outside number by every term inside the brackets. So 3 $x + 4$ becomes 3x + 12, not 3x + 4. Don't forget that second term!

Golden Rule: Always check your answer by substituting it back into the original equation - if both sides equal the same number, you're right!

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Updated Mar 19, 2026

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12 pages

Step-by-Step Guide to Solving Linear Equations in One Variable

When distributing, multiply the outside number by every term inside the brackets. So 3 $x + 4$ becomes 3x + 12, not 3x + 4. Don't forget that second term!

Golden Rule: Always check your answer by substituting it back into the original equation - if both sides equal the same number, you're right!

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