Algebra · Lesson 9 of 16

Solving Equations

An equation is a balance. Do the same thing to both sides, undoing operations in reverse order, until the unknown stands alone.

Secondary
Updated 14 November 2026

What you will learn

Solve one- and two-step equations with inverse operations.
Solve equations with unknowns on both sides and with brackets.
Check answers by substitution.

Before you start: you should be able to simplify expressions and expand brackets. See Expanding Single Brackets.

The idea

An equation says that two expressions are equal, such as $3x + 4 = 19$ . Solving it means finding the value (or values) of the unknown that make the statement true.

Picture a pair of old-fashioned scales. The left pan holds $3x + 4$ , the right pan holds $19$ , and they balance. If you take $4$ off the left, you must take $4$ off the right to keep the balance. That single idea, do the same to both sides, is the whole method.

x + 3 = 7 is a balance. Take 3 from both sides and it stays balanced, leaving x = 4.

Undo in reverse order

Think about what has been done to $x$ . In $3x + 4$ , the $x$ was first multiplied by $3$ , then had $4$ added. To get back to $x$ , undo those steps in reverse: subtract $4$ , then divide by $3$ . It is like taking off your shoes before your socks.

Solving a two-step equation: undo the add, then undo the multiply, then check.

Always check

Substitute your answer back into the original equation. If both sides give the same number, you are right. This takes seconds and catches most errors.

Worked examples

Example 1: A two-step equation

Solve $3x + 4 = 19$ .

Subtract $4$ from both sides: $3x = 15$ .
Divide both sides by $3$ : $x = 5$ .
Check: $3(5) + 4 = 19$ . ✓

Answer: $x = 5$

Example 2: Unknowns on both sides

Solve $7y - 3 = 4y + 12$ .

Collect the $y$ terms on the side with more of them: subtract $4y$ from both sides to get $3y - 3 = 12$ .
Add $3$ to both sides: $3y = 15$ .
Divide by $3$ : $y = 5$ .
Check: left side $7(5) - 3 = 32$ , right side $4(5) + 12 = 32$ . ✓

Answer: $y = 5$

Example 3: Brackets and fractions

Solve $\dfrac{2(n - 1)}{3} = 6$ .

Multiply both sides by $3$ to clear the fraction: $2(n - 1) = 18$ .
Divide both sides by $2$ : $n - 1 = 9$ . (Expanding first also works.)
Add $1$ : $n = 10$ .
Check: $\frac{2(9)}{3} = \frac{18}{3} = 6$ . ✓

Answer: $n = 10$

If the unknown ends up with a negative coefficient, such as $-2x = 8$ , divide by $-2$ to get $x = -4$ . Keep track of the sign.

Why it matters

Equations turn word problems into something you can solve mechanically. How many hours must I work at £11.50 an hour to earn £299? becomes $11.5h = 299$ , so $h = 26$ . Engineers solve equations to size beams, nurses to calculate drip rates, and programmers to position objects on a screen.

The balancing method also scales up. The same logic solves linear equations with graphs, rearranges physics formulas and, with a few extra tools, handles quadratics.

Practice

Your turn. Type or choose an answer, then press Check. Getting one wrong is fine, that is what practice is for. You can try as many times as you like.