By Thread Academy · 20 September 2026 · Mathematics
Knowing what a linear equation looks like is only half the battle. The real skill is finding the value of the unknown that makes the equation true, the number that balances both sides. This guide teaches you the one method that solves them all, then works through every type you will meet at IGCSE: one-step equations, two-step equations, variables on both sides, and brackets.
The golden rule: the balancing method
Think of an equation as a perfectly balanced scale. The left side weighs exactly the same as the right side. If you add, subtract, multiply, or divide the weight on one side, the scale tips, unless you do exactly the same thing to the other side.
That is the whole method: whatever you do to one side, do to the other. Every technique in this article is just an application of that rule. Your aim is always to get the unknown on its own on one side of the equation.
One-step equations
These need a single inverse operation to isolate the unknown.
Worked example 1: an addition equation
Solve x + 7 = 15.
Step 1: The x is trapped by a plus 7, so do the inverse: subtract 7 from both sides.
x + 7 - 7 = 15 - 7
Step 2: Simplify both sides.
x = 8
Worked example 2: a multiplication equation
Solve 4x = 28.
Step 1: The x is trapped by a times 4, so do the inverse: divide both sides by 4.
4x / 4 = 28 / 4
Step 2: Simplify.
x = 7
Two-step equations
These need two inverse operations, and the order matters. Undo addition and subtraction first, then multiplication and division, working from the outside in.
Worked example 3: a two-step equation
Solve 3x + 5 = 20.
Step 1: Undo the addition first. Subtract 5 from both sides.
3x + 5 - 5 = 20 - 5 3x = 15
Step 2: Undo the multiplication. Divide both sides by 3.
x = 5
Notice that we did not divide by 3 first. Dividing first would have given messy fractions. Peeling away the outer layer first keeps the arithmetic clean.
Worked example 4: with a subtraction
Solve 2x - 9 = 11.
Step 1: Add 9 to both sides.
2x = 20
Step 2: Divide both sides by 2.
x = 10
Equations with variables on both sides
When the unknown appears on both sides, gather all the x terms onto one side first, then solve as before. It usually helps to move the smaller x term so you avoid negative coefficients.
Worked example 5: variables on both sides
Solve 5x - 3 = 2x + 9.
Step 1: Subtract 2x from both sides to collect the x terms on the left.
5x - 2x - 3 = 2x - 2x + 9 3x - 3 = 9
Step 2: Add 3 to both sides.
3x = 12
Step 3: Divide both sides by 3.
x = 4
Equations with brackets
Brackets add one preliminary step: deal with the bracket first, either by expanding it or, when the whole bracket is multiplied by a single number, by dividing both sides by that number.
Worked example 6: dividing through the bracket
Solve 2(x + 3) = 16.
Step 1: Both sides can be divided by 2, which removes the bracket in one move.
x + 3 = 8
Step 2: Subtract 3 from both sides.
x = 5
Worked example 7: expanding first
Solve 3(x - 2) + 4 = 19.
Step 1: Expand the bracket by multiplying each term inside by 3. Watch the sign: 3 times -2 is -6.
3x - 6 + 4 = 19
Step 2: Simplify the left side: -6 + 4 is -2.
3x - 2 = 19
Step 3: Add 2 to both sides.
3x = 21
Step 4: Divide both sides by 3.
x = 7
Common mistakes: sign errors
Most wrong answers in this topic come from sign slips, not from misunderstanding the method. Two traps catch students again and again.
Trap 1: forgetting to multiply every term in a bracket. Expanding 2(x - 3) gives 2x - 6, not 2x - 3. Every term inside the bracket must be multiplied, including the constant.
Trap 2: mishandling subtraction when moving terms. Consider 5 - x = 2. A common error is to write x = 7 by adding 5 and 2. The correct working subtracts 5 from both sides:
5 - x - 5 = 2 - 5 -x = -3 x = 3
When a negative sign sits in front of the unknown, isolate the negative term first, then multiply or divide both sides by -1. Slow down on every minus sign and these traps disappear.
Always check your answer
Checking takes ten seconds and catches most slips. Substitute your answer back into the original equation and confirm both sides agree.
Check x = 5 in 3x + 5 = 20:
Left side: 3(5) + 5 = 15 + 5 = 20 Right side: 20
Both sides equal 20, so the answer is correct. Build this habit now. In an exam, a quick check can be the difference between full marks and a lost mark on an otherwise perfect method.
Key takeaways
- The balancing method is the only rule you need: whatever you do to one side of the equation, do to the other.
- In two-step equations, undo addition and subtraction before multiplication and division.
- With variables on both sides, collect the x terms onto one side first, preferably moving the smaller x term.
- With brackets, either divide both sides by the number outside or expand the bracket carefully, multiplying every term inside.
- Sign errors are the most common mistake: double-check every minus sign and every expansion.
- Always verify your answer by substituting it back into the original equation.