10-Week Mathematics Catch-Up Programme (CAPS)

Week 2 – Day 1

Excellent. We've now completed the Grade 8 revision that most learners need before tackling higher algebra. Week 2 begins the transition into Grade 9 mathematics, which is where many Grade 11 learners first begin to struggle if their foundations aren't solid.

The emphasis this week is solving equations, not just simplifying expressions. This is one of the most important skills in the entire CAPS Mathematics curriculum because it appears in functions, coordinate geometry, trigonometry, financial maths, sequences, and calculus.

Solving Linear Equations (One-Step and Two-Step Equations)

Duration: 1 Hour

Grade Level: Grade 9 Foundation

Learning Outcomes

By the end of today's lesson you should be able to:

Understand what an equation is.
Solve one-step equations.
Solve two-step equations.
Check your answers by substitution.
Solve simple word problems involving equations.

Part A – Study Guide

1. What is an Equation?

An equation states that two expressions are equal. It always contains an equals sign (=).

Examples:

x + 5 = 12

3a = 21

2y - 4 = 10

Your goal is to find the value of the unknown (the variable).

2. The Golden Rule of Equations

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

Think of an equation as a balance scale. If you remove or add something to one side, you must do the same to the other side to keep it balanced.

For example:

$x + 5 = 12$

Subtract 5 from both sides:

$x + 5 - 5 = 12 - 5$

$x = 7$

3. Solving One-Step Equations

Example 1

Solve

$x + 8 = 15$

Subtract 8 from both sides:

$x = 7$

Example 2

Solve

$x - 9 = 13$

Add 9 to both sides:

$x = 22$

Example 3

Solve

$5x = 40$

Divide both sides by 5:

$x = 8$

Example 4

Solve

$\frac{x}{6} = 9$

Multiply both sides by 6:

$x = 54$

4. Solving Two-Step Equations

Example 1

Solve

$2x + 3 = 11$

Subtract 3:

$2x = 8$

Divide by 2:

$x = 4$

Example 2

Solve

$3x - 7 = 20$

Add 7:

$3x = 27$

Divide by 3:

$x = 9$

5. Always Check Your Answer

Example

Solve

$4x = 28$

Answer:

$x = 7$

Check:

$4(7)=28$

Correct!

Common Mistakes

❌ Forgetting to do the same operation on both sides.

❌ Dividing before removing the constant.

Example:

$2x+6=14$

Wrong:

14 ÷2 =7

Correct:

Subtract 6 first.

Then divide by 2.

❌ Forgetting negative signs.

Part B – Worked Examples

Example 1

Solve

$x+9=18$

Subtract 9.

Answer:

$x=9$

Example 2

Solve

$x-15=7$

Add 15.

Answer:

Example 3

Solve

$7x=56$

Divide by 7.

Answer:

Example 4

Solve

$\frac{x}{4}=6$

Multiply by 4.

Answer:

Example 5

Solve

$2x+5=17$

Subtract 5.

$2x=12$

Divide by 2.

Answer:

Example 6

Solve

$5x-10=30$

Add 10.

$5x=40$

Divide by 5.

Answer:

Example 7

A number increased by 9 equals 20.

Find the number.

Equation:

$x+9=20$

Answer:

Example 8

Five times a number equals 45.

Find the number.

Equation:

$5x=45$

Answer:

Part C – Guided Practice

Solve the following.

Section A – One-Step Equations

x+6=14

x-11=8

4x=36

\frac{x}{5}=7

x+18=30

x-20=15

9x=81

\frac{x}{8}=9

x+25=60

12x=84

Section B – Two-Step Equations

2x+4=18

3x-6=15

4x+8=36

5x-5=45

6x+12=48

7x-14=35

8x+16=80

9x-9=45

2x+11=25

3x-12=18

Part D – Word Problems

Sarah thinks of a number. She adds 8 and gets 19. What was the number?
Four times a number equals 52. Find the number.
A taxi charges a fixed fee of R20 plus R8 per kilometre. If the total fare is R76, how many kilometres were travelled?
Three times a number minus 5 equals 28. Find the number.
A learner buys 6 identical pens and pays R54. What is the price of one pen?

Challenge Questions

5x+12=47

8x-20=44

The perimeter of a rectangle is 30 cm. Its length is 9 cm. Find its width.

(Hint: Perimeter = 2(length + width))

Twice a number plus 7 equals 31. Find the number.

A mother is 36 years old. She is three times as old as her daughter. How old is the daughter?

Answers

8
19
9
35
12
35
9
72
35
7
7
7
7
10
6
7
8
6
7
10
11
13
7 km
11
R9
7
8
6 cm
12
12 years

Extension Activity (Optional)

For each equation below:

Solve it.
Check your answer by substituting it back into the original equation.

a) $(4x + 9 = 29)$

b) $(7x - 8 = 41)$

c) $(\frac{x}{9} = 11)$

d) $(6x + 5 = 41)$

Parent's Notes

Today's focus is on understanding the process, not just getting the right answer. Encourage the student to:

Write one step per line.
State the operation they are performing (e.g. "Subtract 5 from both sides").
Always check their solution by substituting it back into the original equation.

This habit of checking work becomes invaluable in Grade 10 and Grade 11, where equations become more complex and careless mistakes are more common.

Tomorrow (Week 2 – Day 2) we'll extend these skills by solving equations with brackets and variables on both sides, which are common in Grade 9 and Grade 10 CAPS examinations.