Algebra

Solving Linear Equations

Match Equation to Solution

Draw a line from each equation to its solution.

x + 5 = 12

y − 3 = 10

2n = 16

m ÷ 4 = 5

3k = 21

k = 7

One-Step Equations

Circle the correct solution.

x + 8 = 15

x = 7

x = 23

x = 6

y − 6 = 9

y = 15

y = 3

y = 14

3n = 24

n = 8

n = 21

n = 27

m ÷ 5 = 7

m = 35

m = 12

m = 2

Two-Step Equations

Solve each equation. Circle the correct answer.

2x + 3 = 11

x = 4

x = 7

x = 5

3y − 4 = 14

y = 6

y = 5

y = 10

n ÷ 2 + 5 = 9

n = 8

n = 7

n = 4

4m − 8 = 20

m = 7

m = 3

m = 12

Which Step Comes First?

Sort these steps for solving 3x + 6 = 18 in the correct order.

Divide both sides by 3 → x = 4

Start: 3x + 6 = 18

Subtract 6 from both sides → 3x = 12

Step 1

Step 2

Step 3

Equation — Situation Match

Match each real-life situation to its equation.

I have x chocolates. I give away 4. I have 9 left.

Three equal piles of coins make 24 coins.

Double a number plus 5 equals 17.

A number divided by 6 equals 8.

2n + 5 = 17

3c = 24

x − 4 = 9

n ÷ 6 = 8

Write and Solve Equations

Write an equation for each problem, then solve. Show your steps.

Mia thinks of a number, doubles it, and adds 3. The result is 19. What is the number?

Tickets to a show cost $t each. Four tickets plus a $5 booking fee totals $37. Find t.

A rope is cut into 6 equal pieces each 8 cm long. Write an equation and find the original length.

Check the Solution

Substitute the proposed solution. Is it correct?

2x + 4 = 14, x = 5?

Correct: 2×5+4=14 ✓

Incorrect

Cannot tell

3y − 2 = 10, y = 3?

Incorrect: 3×3−2=7≠10

Correct

Cannot tell

n/4 + 1 = 6, n = 20?

Correct: 20/4+1=6 ✓

Incorrect

Not enough info

5m − 3 = 22, m = 5?

Correct: 5×5−3=22 ✓

Incorrect

Cannot tell

Variables on Both Sides

Solve each equation where the variable appears on both sides.

5x = 3x + 8

x = 4

x = 8

x = 2

4n − 2 = 2n + 6

n = 4

n = 8

n = 2

7y + 1 = 4y + 10

y = 3

y = 9

y = 1

2m + 9 = 5m

m = 3

m = 9

m = 4

TipCollect variable terms on one side by subtracting the smaller variable term from both sides.

Balance-Scale Equations

Each balance scale shows an equation. Match the scale description to its solution.

Left: 2 bags + 6 counters, Right: 18 counters

Left: 3 bags + 4 counters, Right: 19 counters

Left: 4 bags, Right: 2 bags + 10 counters

Left: 5 bags − 3 counters, Right: 12 counters

x = 3

x = 5

x = 6

x = 5

Equations from Diagrams and Situations

Write an equation for each situation and solve it.

A rectangle has perimeter 38 cm. Its length is (2x + 3) cm and width is x cm. Write and solve an equation to find x.

Four friends each pay the same amount. They also use a $12 voucher. Total gift cost $60. Write an equation and find each person's contribution.

Which Equation Matches?

A scale has 3 bags on the left and a 15-coin pile on the right. Sort: which equations correctly model this?

3x = 15

x + 3 = 15

x = 15 ÷ 3

3 + x = 5

15 = 3 × x

Correct equation

Incorrect equation

Real-World Equation Problems

Write and solve an equation for each problem.

A plumber charges a $60 call-out fee plus $45 per hour. The total bill was $195. How many hours did the plumber work?

Two friends are saving. Ava has $85 and saves $12 per week. Ben has $50 and saves $18 per week. Write and solve an equation for when they have equal amounts.

Equations with Brackets

Solve each equation. Circle the correct answer.

2(x + 5) = 18

x = 4

x = 9

x = 6

3(2y − 1) = 15

y = 3

y = 2

y = 6

4(n − 3) + 2 = 10

n = 5

n = 6

n = 7

5(2m + 3) = 25

m = 1

m = 4

m = 3

Solving Equations — Show Full Working

Solve each equation, writing a clear step at each line.

5(x − 3) = 2x + 6

4(2n + 1) = 3(n + 8)

Geometry and Equations

Write and solve an equation to find the unknown angle or side.

A triangle has angles (2x + 10)°, (x + 20)°, and (3x)°. Write and solve an equation to find x, then find each angle.

Angles on a straight line: (4y + 10)° and (2y − 4)°. Write and solve an equation to find y.

TipIn a triangle, angles sum to 180°. In a straight line, angles sum to 180°.

Sort by Number of Steps to Solve

Sort each equation by how many steps are needed to solve it.

x + 4 = 10

2x + 3 = 11

3(x − 2) + 4 = 13

5n = 35

4y − 7 = 2y + 9

n/3 = 8

1 step

2 steps

3+ steps

TipA one-step equation has one operation to undo. A two-step equation has two operations to undo.

Equation Puzzles — Number Tricks

Solve each mystery number puzzle using algebra.

I triple a number, subtract 8, and double the result. The answer is 22. Find the number.

When a number is added to three times itself, the result is 44. Find the number.

Equations — Communicate Your Reasoning

Explain each step as you solve.

Solve 7x − 4 = 3x + 16. Write the name of the operation at each step (e.g., 'subtract 3x from both sides').

A friend says x = 2 is the solution to 5x + 3 = 10. Show clearly why they are wrong and find the correct solution.

Simultaneous Equations Preview

Solve this pair of equations by trial and by substitution.

x + y = 10 and x − y = 4. By trial, find the values of x and y.

Verify your solution by substituting back into both equations.

TipSimultaneous equations are two equations with two unknowns. You need to find values that satisfy both at once.

Real-World Multi-Step Equations

Model and solve each real-world problem.

A rectangle's length is 3 times its width. The perimeter is 64 cm. Find the dimensions.

A cinema charges $12 for adults and $8 for children. A group of 6 people paid $56. Write and solve an equation to find how many adults and children there were.

Equation Solving Investigation

Explore what happens in each case.

Solve: x² = 9. How many solutions are there? What are they?

Solve: x² = −4. What do you find? Explain.

Solve: |x| = 5. How many solutions?

Rearranging Real Formulas

Rearrange each formula and use it.

The formula for speed is v = d/t. Rearrange to find d in terms of v and t. Find d when v = 80 km/h and t = 2.5 hours.

C = 5(F − 32)/9. Rearrange to find F in terms of C. Convert 100°C to Fahrenheit.

Design an Equation Activity

Create a set of equation problems for a peer.

Write a one-step equation problem with a real-world context. Include the solution.

Write a two-step equation problem with a real-world context. Include the solution.

Write a variables-on-both-sides problem. Include the solution.

Extend: Equations with Surds

Solve each equation involving square roots.

√x = 7. Find x.

√(2x + 1) = 5. Find x.

x² − 5 = 44. Find both values of x.

Linear Equations in Two Variables

Explore the solution set of a linear equation in two variables.

For the equation x + y = 10, find 5 pairs (x, y) that satisfy it. Plot them on a coordinate grid.

Draw here

Describe the shape made by all solutions. What type of mathematical object is it?

Project: The Equation and the Graph

Connect algebraic solutions to graphical representations.

Plot y = 2x − 4 for x from −2 to 4. Where does the line cross the x-axis (i.e., where y = 0)? How does this connect to solving 2x − 4 = 0?

Draw here

Plot y = x + 1 on the same grid. Where do the two lines meet? What equation does this intersection solve?

TipEvery linear equation y = mx + c has a straight-line graph. The solution of an equation is where the graph crosses the x-axis.

Reflection: Solving Equations

Summarise your learning about linear equations.

Explain in your own words why we 'do the same to both sides' when solving an equation.

Describe three real-world situations where you would need to solve an equation. For each, write the equation.

What is the hardest type of equation you can now solve? Solve a fresh example of it.

Equation Scavenger Hunt

Find and write equations in real life.

1Look at a food packet. If one serve has x kilojoules and there are 4 serves, write an equation for total kilojoules and solve for x.
2Plan a shopping trip: you have $50 and each item costs the same amount. If you can buy exactly 4 items with $2 change, write and solve the equation.
3Measure a room and write an equation for perimeter in terms of length l and width w. Solve for an unknown dimension.
4Create a mystery number puzzle for a family member and have them solve it.
5Look up 'simultaneous equations' online and read about how they are used to solve real business problems.

Extending — Inequalities

Inequalities are like equations, but use <, >, ≤, or ≥ instead of =.

Solve and show on a number line: 2x + 3 > 9

Solve: −3x ≤ 12. Explain why the sign flips when dividing by −3.

Write a real-world situation that could be represented by the inequality 5x + 20 ≤ 100. Solve it and interpret the answer.

TipKey rule: if you multiply or divide both sides by a negative number, you must flip the inequality sign.

Extending — Verify Solutions

Circle Yes or No: does the given value satisfy the equation?

x = 4 in 3x − 2 = 10?

Yes

x = −2 in x² − 4 = 0?

Yes

x = 5 in 2(x + 1) = 14?

Yes

x = 3 in 4x/3 = 4?

Yes

Extending — Setting Up Equations from Geometry

Write and solve an equation for each geometric problem.

A triangle has angles in the ratio 1:2:3. Write an equation using the angle sum property and find each angle.

Two supplementary angles differ by 30°. Let the smaller angle be x°. Write and solve the equation.

A rectangle has perimeter 96 cm. Its length is (3x + 2) cm and its width is (x + 6) cm. Solve for x and find the dimensions.

Extending — Simultaneous Equations Introduction

A pair of equations with two unknowns — solve them together.

Two numbers add to 20 and one is 4 more than the other. Let them be x and y. Write two equations, then find x and y by substitution.

A shop sells adult tickets for $12 and child tickets for $8. A group buys 10 tickets and pays $104 total. How many of each type were bought? Set up two equations and solve.

TipSimultaneous equations are formally studied in Year 8, but this introduction builds important foundations.

Extending — Match Word Problem to Equation

Draw a line to match each word problem to the equation that represents it.

A number doubled then decreased by 5 equals 11

Three times a number plus 4 equals twice the number plus 9

The sum of four consecutive integers is 58

A quarter of a number is 3 more than 5

n + (n+1) + (n+2) + (n+3) = 58

2n − 5 = 11

3n + 4 = 2n + 9

n/4 = 8

Extending — Error Analysis

A student made errors solving the following equations. Find and correct each mistake.

Student writes: 3x + 6 = 15 → 3x = 15 − 6 → 3x = 9 → x = 27. What is the error? Write the correct solution.

Student writes: 2(x + 3) = 14 → 2x + 3 = 14 → 2x = 11 → x = 5.5. What is the error? Write the correct solution.

Student writes: x/4 + 2 = 5 → x/4 = 7 → x = 7/4. What is the error? Write the correct solution.

What to do next

Algebra

Tables of Values & Variable Relationships

Create tables of values from algebraic expressions and describe how variables change