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Solving Equations

What you'll learn

  • What an equation is and what it means to solve one.
  • How to use inverse operations to find an unknown.
  • How to solve equations with brackets, fractions, and negative answers.
  • How to handle equations with the unknown on both sides.

1. Equations are balanced calculations

An equation is like a set of scales: the left side and the right side have the same value.

A balanced scale shows that both sides of an equation must have the same value.

Definition

Equation, unknown and solution

  • An equation is a mathematical statement with an equals sign, such as x+4=11x+4=11x+4=11.
  • The unknown is the value you do not know yet, usually shown by a letter like xxx, yyy, or mmm.
  • A solution is the value of the unknown that makes the equation true.
Key Idea

Keep the balance

Whatever you do to one side of an equation, you must do to the other side as well.

Doing the same operation to both sides keeps the equation balanced.

Example

Finding a missing number

Find the missing number in 14−n=914-n=914−n=9.

A bar model shows that the missing amount n is the difference between 14 and 9.

  1. Think: 14 minus what gives 9?

  2. The difference between 14 and 9 is 5.

  3. So the missing number is n=5n=5n=5.

  4. Check: 14 minus 5 gives 9, so it works.

2. One-step equations: adding and subtracting

An inverse operation is the operation that undoes another operation.

Definition

Inverse operations

  • Adding is undone by subtracting.
  • Subtracting is undone by adding.
  • Multiplying is undone by dividing.
  • Dividing is undone by multiplying.

Example: undoing addition

Example

Solve an equation like x plus a number

Solve x+6=18x+6=18x+6=18.

The bar model represents x plus 6 making a total of 18.

  1. The unknown xxx has had 6 added to it.

  2. Undo adding 6 by subtracting 6 from both sides: x=18−6x=18-6x=18−6.

  3. So x=12x=12x=12.

Example: undoing subtraction

Example

Solve an equation like x minus a number

Solve p−7=11p-7=11p−7=11.

  1. The unknown ppp has had 7 subtracted from it.

  2. Undo subtracting 7 by adding 7 to both sides: p=11+7p=11+7p=11+7.

  3. So p=18p=18p=18.

Common Mistake

Only changing one side

If you subtract 6 from the left side, you must subtract 6 from the right side too. The equals sign means the two sides must stay balanced.

3. One-step equations: multiplying and dividing

When a number is next to a letter, it means multiplication. For example, 7y7y7y means 7 multiplied by yyy.

Definition

Coefficient

The coefficient is the number multiplying the unknown. In 7y7y7y, the coefficient is 7.

Multiplication equations

Example

Undo multiplication by dividing

Solve 8a=568a=568a=56.

Eight equal parts of size a make a total of 56.

  1. 8a8a8a means 8 multiplied by aaa.

  2. Undo multiplying by 8 by dividing both sides by 8.

  3. a=56÷8a=56\div 8a=56÷8.

  4. So a=7a=7a=7.

Division equations

Example

Undo division by multiplying

Solve d2=8.5\frac{d}{2}=8.52d​=8.5.

  1. d2\frac{d}{2}2d​ means ddd divided by 2.

  2. Undo dividing by 2 by multiplying both sides by 2.

  3. d=8.5×2d=8.5\times 2d=8.5×2.

  4. So d=17d=17d=17.

Tip

Fraction bar means divide

In equations, a fraction line is another way of showing division. For example, y3\frac{y}{3}3y​ means yyy divided by 3.

4. Equations can have negative answers

A negative number is a number less than zero, such as -3 or -10.

Sometimes the answer to an equation is negative. This is normal.

Example

Solving when the unknown is being subtracted

Solve 6−m=146-m=146−m=14.

A number line helps show that subtracting a negative moves to the right from 6 to 14.

  1. Since 6 minus something gives a bigger answer, the unknown must be negative.

  2. Subtract 6 from both sides: −m=8-m=8−m=8.

  3. If −m=8-m=8−m=8, then m=−8m=-8m=−8.

  4. Check: 6 minus -8 is 14, so m=−8m=-8m=−8 is correct.

Common Mistake

Forgetting the minus sign

If you get −m=8-m=8−m=8, the answer is not m=8m=8m=8. The answer is m=−8m=-8m=−8.

5. Two-step equations

A two-step equation needs two inverse operations to solve it.

Usually, undo adding or subtracting first, then undo multiplying or dividing.

Key Idea

Work backwards

Solve equations by undoing operations in the reverse order to how they were done.

Example

A two-step equation

Solve 3h−5=163h-5=163h−5=16.

A function-machine diagram shows the operations on h and the reverse operations used to solve.

  1. The unknown has been multiplied by 3, then 5 has been subtracted.

  2. Undo subtracting 5 by adding 5 to both sides: 3h=213h=213h=21.

  3. Undo multiplying by 3 by dividing both sides by 3.

  4. So h=7h=7h=7.

Example

A two-step equation with division

Solve y3−4=6\frac{y}{3}-4=63y​−4=6.

  1. Undo subtracting 4 by adding 4 to both sides: y3=10\frac{y}{3}=103y​=10.

  2. Undo dividing by 3 by multiplying both sides by 3.

  3. So y=30y=30y=30.

6. Equations with brackets

Brackets group parts of an expression together. For example, in 4(x−3)4(x-3)4(x−3), the whole bracket is being multiplied by 4.

You can often solve these by undoing the outside operation first.

Example

A bracket equation

Solve 5(x−4)=355(x-4)=355(x−4)=35.

  1. The whole bracket is multiplied by 5.

  2. Undo multiplying by 5 by dividing both sides by 5: x−4=7x-4=7x−4=7.

  3. Undo subtracting 4 by adding 4 to both sides.

  4. So x=11x=11x=11.

Example

A bracket with 2x inside

Solve 20=4(2x−1)20=4(2x-1)20=4(2x−1).

  1. The whole bracket is multiplied by 4, so divide both sides by 4: 5=2x−15=2x-15=2x−1.

  2. Undo subtracting 1 by adding 1 to both sides: 6=2x6=2x6=2x.

  3. Undo multiplying by 2 by dividing both sides by 2.

  4. So x=3x=3x=3.

Tip

Divide the whole bracket first

If an equation looks like 4(x−2)=284(x-2)=284(x−2)=28, it is often easiest to divide both sides by 4 before dealing with the inside of the bracket.

7. Fractions with an expression on top

The top of a fraction is called the numerator. The bottom is called the denominator.

If the whole numerator is divided by a number, multiply both sides by that number first.

Example

A fraction equation

Solve d+54=6\frac{d+5}{4}=64d+5​=6.

  1. The whole top, d+5d+5d+5, is divided by 4.

  2. Undo dividing by 4 by multiplying both sides by 4: d+5=24d+5=24d+5=24.

  3. Undo adding 5 by subtracting 5 from both sides.

  4. So d=19d=19d=19.

Example

A fraction with 2x on top

Solve 2x3=8\frac{2x}{3}=832x​=8.

  1. Undo dividing by 3 by multiplying both sides by 3: 2x=242x=242x=24.

  2. Undo multiplying by 2 by dividing both sides by 2.

  3. So x=12x=12x=12.

8. Unknowns on both sides

Sometimes the unknown appears on both sides of the equals sign.

A term is a part of an expression, such as 5x5x5x, −2x-2x−2x, or 7. Like terms have the same letter part, such as 6x6x6x and 2x2x2x.

The aim is to collect the unknown terms on one side and the number terms on the other.

Example

Unknown on both sides

Solve 6w+9=3w+246w+9=3w+246w+9=3w+24.

A balance diagram shows unknown terms and number terms on both sides before collecting like terms.

  1. Subtract 3w3w3w from both sides to get the unknowns on one side: 3w+9=243w+9=243w+9=24.

  2. Subtract 9 from both sides: 3w=153w=153w=15.

  3. Divide both sides by 3.

  4. So w=5w=5w=5.

Tip

Try to keep the unknown positive

If possible, move the smaller unknown term to the other side. This can help you avoid negative unknowns.

Exam technique

In the exam

  1. Write one clear line for each inverse operation you do.

  2. Do the same operation to both sides every time.

  3. Substitute your answer back into the original equation to check it works.

Self review

Check yourself

  • What operation undoes adding 9?

  • If −m=12-m=12−m=12, what is mmm?

  • When the unknown appears on both sides, which terms should you collect together first?

Recap questions

1 of 5

Solve x+7=19x+7=19x+7=19.

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Lesson

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8 minute activity

Start lesson

Balance scale with a box labelled x and four unit blocks on the left, eleven unit blocks on the right, and arrows showing subtract 4 from both sides to get x = 7

An equation says two expressions have the same value, so it helps to picture it as balanced scales. The letter is the unknown, and solving the equation means finding the value that makes the statement true.

To keep the balance, do the same operation to both sides. These matching actions are usually inverse operations, such as subtracting to undo adding or dividing to undo multiplying.

A value is a solution only if it works when you substitute it back into the original equation. Checking at the end helps you catch sign mistakes and arithmetic slips.

Flashcards

Remember key concepts with flashcards

22 flashcards

Practice flashcards

In an equation, the left and right sides must have the [     ].

Solving Equations Revision Guide

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