Solving Equations with Brackets

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Solving Equations with Brackets

Open with the idea that an equation is like a balance: both sides must stay equal. This will help students make sense of undoing operations on both sides later.

Learning goal

I can solve equations with brackets and pronumerals on both sides, including two-step equations, and check my answers using my calculator.

What words come to mind when you hear solving equations?

Use this to surface prior knowledge such as balance, brackets, x, solve, check, calculator, both sides, and inverse. Invite a few students to explain one word they added.

Two-step equations

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

• First undo addition or subtraction

• Then undo multiplication or division

• Do the same operation to both sides

• Keep the pronumeral by itself

Use simple examples orally, such as x + 5 = 12 and 3x = 18, before combining them into one full two-step example.

Solving a two-step equation

Example: 3x + 4 = 19

• Subtract 4 from both sides

• 3x = 15

• Divide both sides by 3

• x = 5

• Check: 3(5) + 4 = 19

Emphasise the order of undoing: remove the +4 before dividing by 3. Students often try to divide first.

Why checking matters

A solution is correct only if it makes the equation true.

• Substitute the value back in

• Work out both sides

• If both sides match, the solution is correct

Link checking to the meaning of an equation, not just to getting a mark. A wrong answer often shows up immediately when substituted back in.

What should you undo first in 3x + 4 = 19?

A

Add 4

B

Subtract 4

C

Divide by 3

D

Multiply by 3

Tests the key idea from the two-step example: undo addition before division. A common error is dividing first.

Complete the method

A two-step equation needs two inverse steps. First undo or subtraction. Then undo multiplication or . Do the same operation to sides. At the end, the value back in to check it.

A two-step equation needs two inverse steps. First undo

or subtraction. Then undo multiplication or

. Do the same operation to

sides. At the end,

addition

double

one

expand

Correct blanks are addition, division, both, and substitute. This checks the method and the idea of checking answers.

Brackets first

When an equation has brackets, expand them before solving.

• Multiply the term outside by each term inside

• Example: 2(x + 3) = 14

• Expand: 2x + 6 = 14

• Then solve the equation as usual

Remind students that 2(x + 3) means 2 multiplied by everything inside the bracket, not just the first term.

Expanding brackets carefully

Example: 3(a - 2) = 15

• Expand: 3a - 6 = 15

• Add 6 to both sides

• 3a = 21

• Divide by 3

• a = 7

Point out that the negative sign stays with the 2 when expanding. This is a common place for sign errors.

What is 3(a - 2) after expanding?

A

3a - 2

B

3a + 6

C

3a - 6

D

a - 6

The distractors target common errors: not multiplying both terms, losing the negative sign, or dropping the coefficient.

Complete the worked example

When an equation has brackets, them first. In 2(x + 3) = 14, the equation becomes = 14. Then solve as usual. In 3(a - 2) = 15, adding 6 gives = 21, so a = .

When an equation has brackets,

them first. In 2(x + 3) = 14, the equation becomes

= 14. Then solve as usual. In 3(a - 2) = 15, adding 6 gives

= 21, so a =

factor

2x + 3

3a - 21

5

Correct blanks are expand, 2x + 6, 3a, and 7. This anchors bracket expansion before solving.

Pronumerals on both sides

Some equations have the pronumeral on both sides.

• Move the pronumeral terms to one side

• Move the number terms to the other side

• Use inverse operations step by step

• Keep the equation balanced

Students may ask why we 'move' terms. Explain that moving really means doing the same operation to both sides.

Collecting pronumerals on one side

Example: 5x + 2 = 2x + 11

• Subtract 2x from both sides

• 3x + 2 = 11

• Subtract 2 from both sides

• 3x = 9

• x = 3

Stress that subtracting 2x from both sides keeps the equation balanced and reduces the number of pronumeral terms.

Brackets and pronumerals on both sides

Example: 2(x + 1) = x + 7

• Expand: 2x + 2 = x + 7

• Subtract x from both sides

• x + 2 = 7

• Subtract 2

• x = 5

This slide joins the two ideas in the lesson: first expand the bracket, then gather pronumerals onto one side.

Using my calculator

A calculator can help you check your work.

• Type each side separately

• Use brackets exactly as written

• Substitute your answer for the pronumeral

• Compare the two results

Avoid presenting the calculator as a replacement for algebra steps. Its main role here is checking substitution and arithmetic accuracy.

Calculator check example

For 2(x + 1) = x + 7 and x = 5:

• Left side: 2(5 + 1) = 12

• Right side: 5 + 7 = 12

• Both sides are equal

• So x = 5 is correct

If useful, model how incorrect bracket entry on a calculator can give the wrong result, for example typing 2 × 5 + 1 instead of 2 × (5 + 1).

Common mistakes to avoid

• Forgetting to expand every term in a bracket

• Doing different operations on each side

• Moving terms without keeping the balance

• Making sign mistakes with subtraction

• Not checking the final answer

Use these as quick reminders while reviewing worked examples. Sign errors and incomplete expansion are the most common problems at this level.

What should you do first in 5x + 2 = 2x + 11?

A

Subtract 2

B

Add 11

C

Subtract 2x

D

Divide by 5

Checks that students collect pronumerals on one side first rather than jumping to number moves or division.

Complete the solving steps

Some equations have the pronumeral on sides. In 2(x + 1) = x + 7, first the bracket to get 2x + 2 = x + 7. Then subtract from both sides, subtract 2, and get x = .

Some equations have the pronumeral on

sides. In 2(x + 1) = x + 7, first

the bracket to get 2x + 2 = x + 7. Then subtract

from both sides, subtract 2, and get x =

one

factorise

7

2

Correct blanks are both, expand, x, and 5. This links the two main ideas: brackets first, then collect pronumerals.

How can a calculator help you check an answer?

Expected answer elements: students should mention typing each side separately, using brackets exactly as written, substituting the answer for the pronumeral, and checking that both sides match.

What is one thing you learned today?

Encourage brief reflections such as brackets first, both sides stay balanced, checking by substitution, or collecting pronumerals on one side.

Which part felt hardest for you?

Useful responses may mention expanding brackets, sign mistakes, deciding what to do first, or keeping the equation balanced.

Glossary

equation - a maths statement showing two expressions are equal

pronumeral - a letter used to stand for an unknown number

inverse operation - an operation that undoes another operation

expand - multiply a term outside a bracket by each term inside

substitute - replace a pronumeral with a number

solution - the value that makes an equation true

balanced - equal on both sides of the equation