Linear Equations
Solving equations with one unknown, including brackets.
Learn
Linear Equations, explained point by point
Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.
Solve one- and two-step equations
Undo what has been done to x, in reverse order, doing the same to both sides. For 3x + 5 = 20: subtract 5 from both sides (3x = 15), then divide by 3 (x = 5). Every step keeps the equation balanced - that is the whole game.
Solve equations with brackets
Expand the bracket first, then solve as normal: 4(x − 2) = 20 becomes 4x − 8 = 20, so 4x = 28 and x = 7. If everything on the other side divides neatly, you can divide first instead: 4(x − 2) = 20 gives x − 2 = 5 straight away.
Solve equations with the unknown on both sides
Collect the x terms on one side and the numbers on the other. For 5x − 4 = 2x + 11: subtract 2x from both sides (3x − 4 = 11), add 4 (3x = 15), divide (x = 5). Moving the smaller x term keeps the coefficient positive and the signs friendly.
Solve equations involving fractions
Clear the fractions by multiplying every term by the denominator (or the lowest common denominator). For = 5, multiply both sides by 4 to get x + 3 = 20, so x = 17. With two fractions, multiplying by both bottoms - or cross-multiplying - does the same job.
Set up an equation from a worded problem or a diagram, then solve it
Name the unknown, translate the story into symbols, then solve. "I think of a number, double it, add 7, and get 31" becomes 2x + 7 = 31. Angle and perimeter diagrams work the same way - the shape gives you the total, the expressions give you the left-hand side.
Rearrange a formula to change the subject
Use the same inverse-operation moves as equation solving, but the answer is an expression rather than a number. Making u the subject of v = u + at: subtract at, giving u = v − at. If the new subject appears twice, collect its terms and factorise it out first.
Practice
Try a Linear Equations question
A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.
Solve 4(x − 2) = 2x + 10.
- x = 3
- x = 9
- x = 4.5
- x = 1
Show the answer and the working
Answer: x = 9
Expanding the bracket to 4x − 8 lets you collect the x terms on one side and the numbers on the other.
- Expand the bracket: 4x − 8 = 2x + 10.
- Subtract 2x from both sides: 2x − 8 = 10.
- Add 8 to both sides: 2x = 18.
- Divide by 2: x = 9.