How to Solve Linear Equations: The Balance Method

The balance method for one-step, two-step and multi-step equations, why you undo operations in reverse order, how to deal with variables on both sides, and what it means when an equation has no solution or infinitely many.

3 min read Supports 4 lessons

An equation says two things are equal. Picture a pair of scales in balance. If you add something to one pan, you must add the same to the other to keep it level. Solving an equation is just a series of fair moves that end with $x$ alone on one side.

x + 5

12

Take 5 from both sides to keep it level.

Inverse operations

Every operation has an inverse that undoes it. To remove something from beside $x$ , apply its inverse to both sides.

Inverse operations

+ \;\leftrightarrow\; - \qquad\qquad \times \;\leftrightarrow\; \div

Worked example

Solve $x + 5 = 12$ .

1
Subtract $5$ from both sides.
$x = 7$

Tip

Always check by substituting back: $7 + 5 = 12$ ✓. It takes seconds and catches most slips.

Two steps: undo in reverse order

To build $3x - 4$ from $x$ , you multiply by $3$ and then subtract $4$ . To unbuild it, reverse the steps, like taking off shoes before socks: add $4$ first, then divide by $3$ .

$3x - 4 = 11$
The last thing done to $x$ was subtracting 4, so undo that first.

1/5

Variables on both sides

For longer equations, follow a plan: simplify each side (expand brackets, collect like terms), gather the $x$ terms on one side and the numbers on the other, then divide.

Worked example

Solve $5x - 3 = 2x + 9$ .

1
Subtract $2x$ from both sides.
$3x - 3 = 9$

Worked example

Solve $2(x + 4) = 3(x - 1)$ .

1
Expand.
$2x + 8 = 3x - 3$

Tip

Move the $x$ terms towards whichever side already has more of them. That way the coefficient stays positive.

Fractions in equations

If $x$ is divided by something, multiply both sides by it. With several fractions, multiply every term by a common denominator to clear them all at once.

Worked example

Solve $\frac{x}{2} + 7 = 3$ .

1
Subtract $7$ .
$\frac{x}{2} = -4$

No solution, or every solution

Key idea

If the $x$ terms cancel and you are left with something false, like $3 = 7$ , there is no solution. If you are left with something always true, like $3 = 3$ , every value of $x$ works.

Watch out

Whatever you do, do it to the whole of each side. Dividing only one term by $3$ unbalances the scales.

Common questions

What is the balance method?: Treat the equation like balanced scales: whatever you do to one side, do to the other. Use inverse operations, step by step, until the variable is on its own.
Which step do I undo first?: Undo operations in the reverse order to how they were applied. In $3x - 4 = 11$ , $x$ was multiplied by $3$ and then $4$ was subtracted, so add $4$ first and then divide by $3$ .

Practise solving linear equations

The One-Step Equations lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

What is the balance method?

Treat the equation like balanced scales: whatever you do to one side, do to the other. Use inverse operations, step by step, until the variable is on its own.

Which step do I undo first?

Undo operations in the reverse order to how they were applied. In $3x - 4 = 11$ , $x$ was multiplied by $3$ and then $4$ was subtracted, so add $4$ first and then divide by $3$ .