10-Week Mathematics Catch-Up Programme (CAPS)

Week 2 – Day 2

This lesson is where many learners begin to gain real confidence. Up to now, equations have been fairly straightforward. Today she'll learn to remove brackets first and solve equations where the variable appears on both sides. These are very common in Grade 10 and Grade 11 tests.

Solving Equations with Brackets and Variables on Both Sides

Duration: 1 Hour

Grade Level: Grade 9 Foundation

Learning Outcomes

By the end of this lesson you should be able to:

Part A – Study Guide

Step 1: Expand the Brackets

Before solving an equation with brackets, remove the brackets by multiplying every term inside by the number outside.

Example

Expand:

3(x+4)3(x+4)

Multiply each term by 3:

3(x+4)=3x+123(x+4)=3x+12

Example

Expand:

5(2x−3)5(2x-3)

Multiply:

5×2x=10x5 \times 2x = 10x 5×(−3)=−155 \times (-3) = -15

Answer:

10x−1510x-15

Step 2: Collect Like Terms

After removing brackets, move all the variable terms to one side and the numbers to the other.

Solving Equations with Brackets

Example 1

Solve

2(x+3)=142(x+3)=14

Expand:

2x+6=142x+6=14

Subtract 6 from both sides:

2x=82x=8

Divide by 2:

x=4x=4

Check

Substitute (x=4):

2(4+3)=2(7)=142(4+3)=2(7)=14

Correct.

Example 2

Solve

3(x−2)=153(x-2)=15

Expand:

3x−6=153x-6=15

Add 6:

3x=213x=21

Divide by 3:

x=7x=7

Solving Equations with Variables on Both Sides

Example 1

Solve

5x+4=3x+185x+4=3x+18

Subtract (3x) from both sides:

2x+4=182x+4=18

Subtract 4:

2x=142x=14

Divide by 2:

x=7x=7

Example 2

Solve

7x−5=4x+167x-5=4x+16

Subtract (4x):

3x−5=163x-5=16

Add 5:

3x=213x=21

Divide by 3:

x=7x=7

Strategy Checklist

Whenever solving an equation:

✔ Remove brackets first.

✔ Collect variable terms on one side.

✔ Move numbers to the other side.

✔ Divide by the coefficient.

✔ Check your answer.

Common Mistakes

Mistake 1

4(x+5)=4x+54(x+5)=4x+5

❌ Wrong

Correct:

4x+204x+20

Mistake 2

Moving a term across the equals sign without changing the operation.

Example

x+8=20x+8=20

To move 8:

Subtract 8 from both sides.

Don't simply "move it" without showing the operation.

Mistake 3

Stopping too early.

Example

2x=142x=14

Some learners stop here.

Remember:

Divide by 2.

x=7x=7

Part B – Worked Examples

Example 1

Solve

4(x+2)=244(x+2)=24

Expand:

4x+8=244x+8=24

Subtract 8:

4x=164x=16

Divide by 4:

x=4x=4

Example 2

Solve

5(x−1)=305(x-1)=30

Expand:

5x−5=305x-5=30

Add 5:

5x=355x=35

Divide:

x=7x=7

Example 3

Solve

2x+9=x+172x+9=x+17

Subtract (x):

x+9=17x+9=17

Subtract 9:

x=8x=8

Example 4

Solve

6x−8=2x+206x-8=2x+20

Subtract (2x):

4x−8=204x-8=20

Add 8:

4x=284x=28

Divide:

x=7x=7

Example 5

Solve

3(2x+1)=213(2x+1)=21

Expand:

6x+3=216x+3=21

Subtract 3:

6x=186x=18

Divide:

x=3x=3

Example 6

Solve

8+4x=2x+208+4x=2x+20

Subtract (2x):

8+2x=208+2x=20

Subtract 8:

2x=122x=12

Divide:

x=6x=6

Part C – Guided Practice

Section A – Expand First, Then Solve

2(x+5)=182(x+5)=18
3(x−4)=93(x-4)=9
5(x+2)=405(x+2)=40
6(x−3)=246(x-3)=24
4(2x+1)=364(2x+1)=36
2(3x−2)=162(3x-2)=16
7(x+1)=567(x+1)=56
5(2x−4)=205(2x-4)=20
3(x+7)=363(x+7)=36
8(x−2)=328(x-2)=32

Section B – Variables on Both Sides

3x+8=x+183x+8=x+18
6x−4=2x+206x-4=2x+20
5x+12=2x+245x+12=2x+24
8x−6=5x+158x-6=5x+15
9x+4=6x+199x+4=6x+19
7x−9=3x+197x-9=3x+19
10x+5=8x+1910x+5=8x+19
12x−8=9x+1312x-8=9x+13
4x+11=2x+214x+11=2x+21
15x−10=10x+2015x-10=10x+20

Section C – Mixed Questions

4(x+6)=404(x+6)=40
3x+15=x+253x+15=x+25
5(2x+3)=655(2x+3)=65
2x+18=x+252x+18=x+25
7(x−2)=357(x-2)=35

Part D – Word Problems

  1. Three times a number plus 4 equals 25. Find the number.

  2. Twice a number minus 7 equals 17.

  3. Sipho buys five identical notebooks and pays R85. Write an equation and find the price of one notebook.

  4. A rectangle has a length of (x+3) cm and a width of (x) cm. If the perimeter is 26 cm, find (x).

Hint: Perimeter = (2(length+width))(2(\text{length}+\text{width}))

  1. A father is four times as old as his son. Together they are 50 years old. How old is the son?

Challenge Questions

4(2x+3)=5x+184(2x+3)=5x+18
3(3x−2)=2x+253(3x-2)=2x+25

The sum of three consecutive numbers is 48. Find the numbers.

Hint: Let the first number be (x).

A taxi charges a fixed fee of R30 plus R12 per kilometre. If the total fare is R126, how many kilometres were travelled?

A school buys boxes of pencils. Each box contains the same number of pencils. Four boxes plus 12 extra pencils gives a total of 100 pencils. How many pencils are in each box?

Answers

Section A

  1. (x=4)

  2. (x=7)

  3. (x=6)

  4. (x=7)

  5. (x=4)

  6. (x=3)

  7. (x=7)

  8. (x=4)

  9. (x=5)

  10. (x=6)

Section B

  1. (x=5)

  2. (x=6)

  3. (x=4)

  4. (x=7)

  5. (x=5)

  6. (x=7)

  7. (x=7)

  8. (x=7)

  9. (x=5)

  10. (x=6)

Section C

  1. (x=4)

  2. (x=5)

  3. (x=5)

  4. (x=7)

  5. (x=7)

Section D

  1. (x=7)

  2. (x=12)

  3. Equation: (5x=85), so R17 per notebook.

2[(x+3)+x]=262[(x+3)+x]=26 4x+6=264x+6=26 4x=204x=20 x=5x=5
  1. Let the son's age be (x).

Father's age = (4x)

x+4x=50x+4x=50 5x=505x=50 x=10x=10

Son = 10 years, Father = 40 years.

Challenge

  1. (x=2)

  2. (x=317=437)(x=\frac{31}{7}=4\frac{3}{7})

  3. 15, 16, 17

  4. Equation: (30+12x=126)

12x=9612x=96 x=8x=8

8 km

Equation:

4x+12=1004x+12=100 4x=884x=88 x=22x=22

22 pencils per box

Parent's Notes

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

Tomorrow's lesson (Week 2 – Day 3) will introduce inequalities (using (<), (>), (≤\le), and (≥\ge)) and basic formula rearrangement (making a different variable the subject)—two essential skills that feature regularly in Grade 10 and Grade 11 CAPS Mathematics.