Solving Equations

In short: Solving an equation means finding the number the letter stands for. Do the same thing to both sides to keep it balanced, and undo the steps in reverse order: if $3 x + 2 = 14$ , take away 2 and then divide by 3 to get $x = 4$ .

Before you start

An equation has an equals sign, and both sides have the same value.
Inverse operations undo each other: $+$ undoes $-$ , and $\times$ undoes $\div$ . Squaring is undone by a square root.
Square numbers: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100$ . Cube numbers: $1, 8, 27, 64, 125$ .

Keep it balanced

Five bags plus 3 kg balance 18 kg. Take 3 kg off both sides to keep it level.

Equations with brackets

You can expand first, or divide both sides by the number outside. Dividing is often quicker.

Fractions, decimals and squares

In this book the answers to square equations are positive numbers.

Solving equations

Balance

Same thing to both sides
Get the letter alone

Undo

Reverse order
+ undoes -, x undoes ÷

Brackets

Divide by the outside number
Or expand first

Squares and cubes

$a^{2} = 16$ gives $a = 4$
$b^{3} = 125$ gives $b = 5$

Check

Put the answer back in

Quick summary

Do the same thing to both sides to keep an equation balanced.
Undo operations in reverse order: deal with $+$ and $-$ first, then $\times$ and $\div$ .
With brackets, divide by the outside number first, or expand.
For a fraction, multiply both sides by the bottom number.
If $a^{2} = 16$ then $a = 4$ . If $b^{3} = 125$ then $b = 5$ .
Always check by putting your answer back in.

Quiz

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Try 3 questions now

Solve $3 a - 1 = 14$ .
1. A $a = 4$
2. B $a = 5$
3. C $a = 13$
4. D $a = 15$
Show answer
B: $a = 5$
Add 1: $3 a = 15$ . Divide by 3: $a = 5$ .
Solve $15 = 4 a + 3$ .
1. A $a = 4.5$
2. B $a = 3$
3. C $a = 12$
4. D $a = 2$
Show answer
B: $a = 3$
Subtract 3: $12 = 4 a$ . Divide by 4: $a = 3$ .
Solve $4 + 9 x = 31$ .
1. A $x = 3$
2. B $x = 4$
3. C $x = 27$
4. D $x = 35$
Show answer
A: $x = 3$
Subtract 4: $9 x = 27$ . Divide by 9: $x = 3$ .