A linear equation is an equation where the unknown has no power higher than 1, such as 3x + 5 = 20. To solve it, you do the same thing to both sides until x is alone. This page goes from one step to brackets and fractions.
Think of a balance. Keep both sides level.
Hello, curious mind!
What is a linear equation?
A linear equation is an equation in which the unknown appears only to the power 1. It is never squared, never under a square root and never on the bottom of a fraction.
4x − 9 = 3 is linear. x² = 9 is not, because x is squared. 6 ÷ x = 2 is not, because x is on the bottom.
“Linear” comes from the Latin word for “line”. With two letters, such as y = 2x + 1, it makes a straight line on a graph. This page is about one letter and one answer.
Variable
The letter for the number you want.
Coefficient
The number that multiplies x in 4x − 9 = 3.
Constant
A number on its own. Keep its sign.
Solution
Here x = 3, because 12 − 9 = 3.
Remember this
Whatever you do to one side, do to the whole of the other side.
How to solve a linear equation, step by step
Use the balance method. Undo each operation with its opposite: + and − undo each other, and × and ÷ undo each other. Each example ends with a check.
Example 1: one step, x − 8 = 23
The 8 is taken away from x. To undo it, add 8 to both sides: x = 23 + 8 = 31.
Check: 31 − 8 = 23. Both sides match.
Example 2: brackets, 5(x − 3) = 2x + 9
Open the bracket. Multiply both terms inside by 5: 5x − 15 = 2x + 9.
Get the x terms on one side. Take 2x from both sides: 3x − 15 = 9.
Add 15 to both sides: 3x = 24. Divide both sides by 3: x = 8.
Check in the first line, not your last line: 5 × (8 − 3) = 25, and 2 × 8 + 9 = 25.
Example 3: fractions, (x + 1)/2 − (x − 3)/4 = 3
Clear the fractions. The bottoms are 2 and 4, so multiply every term by 4: 2(x + 1) − (x − 3) = 12.
Open the brackets. The minus in front of (x − 3) changes both signs: 2x + 2 − x + 3 = 12.
Collect terms: x + 5 = 12, so x = 7.
Check: 8/2 − 4/4 = 4 − 1 = 3.
Watch out!
In Example 3, writing −x − 3 in place of −x + 3 gives x = 13. The check then fails: 7 − 2.5 is 4.5, not 3.
A linear equation word problem: a Dubai taxi fare
These prices are made up for practice. Real fares vary.
A taxi costs AED 8 to start, plus AED 1.50 for each km. A ride costs AED 32. How far was the ride?
Name the unknown in words: let d be the number of km.
Write the equation: 8 + 1.5d = 32.
Take 8 from both sides: 1.5d = 24. Divide by 1.5: d = 16.
Answer in words, with units: the ride was 16 km. Check: 8 + 1.5 × 16 = 8 + 24 = 32.
For parents
Ask your child what d stands for before they write the equation. Most word-problem errors start there.
Common linear equation mistakes
- Moving a term without changing its sign. From 4x + 7 = 19, some students write 4x = 19 + 7 = 26. The 7 must be taken away: 4x = 12, so x = 3.
- Doing a step to one term only. To clear x/3 + 2 = 6, multiply every term by 3: x + 6 = 18, so x = 12. Writing x + 2 = 18 gives the wrong answer, 16.
- Copying a number wrongly. In the taxi problem, AED 23 in place of AED 32 gives 10 km, not 16 km.
- Slips with negatives. −3(x − 2) is −3x + 6, because −3 × −2 = +6. So −3(x − 2) = 12 gives x = −2, not −6.
- Skipping lines. Two steps done in your head is where signs get lost. Write one step on each line.
Check this with the Free BMI Maths Test
- Moves a term across without changing its sign: CNC, concept not clear.
- Does the step to one side or one term only: CNC, concept not clear.
- Copies a number wrongly from the word problem: CWV, copied a wrong value.
- Slips with negatives, such as −3 × −2: CMT, calculation mistake.
- Skips lines and loses track: BPT, bad presentation.
The Free BMI Maths Test finds which of the 6 mistake types your child makes, and their real maths level. About the Free BMI Maths Test
Test prep corner
Linear equations in SAT maths
College Board lists “Linear equations in 1 variable” as a question type in the Algebra area of SAT maths. These are SAT-style practice questions, not real past questions.
SAT-style: If (2/3)k − 4 = 10, what is the value of k?
Add 4 to both sides: (2/3)k = 14. To undo “× 2/3”, multiply by 3/2: k = 14 × 3/2 = 21. Trap: multiplying 14 by 2/3 gives 28/3. Check: 2/3 of 21 is 14, and 14 − 4 = 10.
SAT-style: A tank holds 900 litres of water. It loses 35 litres each hour. After how many hours does it hold 375 litres?
The equation is 900 − 35h = 375. Take 375 from 900: 35h = 525. Divide by 35: h = 15.
Try a full set of SAT-style maths practice questions, or read the Digital SAT maths guide.
Facts checked on 9 November 2026 on satsuite.collegeboard.org.
Quick check: 5 linear equation questions
1. Which of these is a linear equation?
Not quite. x is squared, so it is not linear.
Yes! x has power 1 and is not on the bottom.
Not quite. x is on the bottom of a fraction.
2. Solve x ÷ 4 = 7.
Yes! Multiply both sides by 4: 7 × 4 = 28.
Not quite. You took 4 away. Undo ÷ with ×.
Not quite. You divided again. Multiply by 4.
3. Solve 6x − 2 = 4x + 10.
Not quite. The −2 moved but kept its sign. 2x = 12.
Not quite. Collect the x terms on one side first.
Yes! 2x − 2 = 10, so 2x = 12 and x = 6.
4. Solve 2(x + 4) = 18.
Not quite. The 2 multiplies the 4 too: 2x + 8.
Yes! x + 4 = 9, so x = 5.
Not quite. Divide 18 by 2 before you take 4 away.
5. Solve 7 − 2x = 15.
Not quite. −2x = 8, so x is negative.
Not quite. You added 7 instead of taking it away.
Yes! −2x = 8, so x = −4. Check: 7 + 8 = 15.
Got 2 or fewer right? Book the Free BMI Maths Test to find the cause.
Questions people ask about linear equations
What makes an equation linear?
The unknown has power 1 only. It is not squared, not under a square root and not on the bottom of a fraction. Its graph is a straight line.
How do you solve a linear equation with fractions?
Multiply every term by a number that all the bottoms divide into. For x/2 + x/3 = 10, multiply by 6: 3x + 2x = 60, so x = 12.
Can a linear equation have no solution?
Yes. 2(x + 3) = 2x + 5 becomes 6 = 5, which is never true, so there is no solution. 3(x − 1) = 3x − 3 is true for every x, so it has infinitely many solutions.
Is y = x² linear?
No. x is squared, so its graph is a curve, not a straight line. That makes it a quadratic.
Do signs and brackets still trip your child up?
The Free BMI Maths Test finds your child’s real maths level and the reason for each mistake.
