Grade 9 Mathematics
Grade 9 · Term 2Mathematics

Algebraic Equations

We solve linear equations using additive and multiplicative inverses, and extend to equations involving fractions, brackets, and exponents. We apply equations to solve real-world word problems systematically.

2.1 Solving Linear Equations2.2 Equations from Word Problems
Weeks 3–4

2.1 Solving Linear Equations

  • Solve linear equations using additive and multiplicative inverses
  • Solve equations with brackets and fractions
  • Solve equations involving laws of exponents
  • Solve equations by factorisation (e.g. x² − 9 = 0 → x = ±3)
🌍

Real-World Connection

Solving an equation is like balancing a perfectly level scale. Whatever you do to one side you MUST do to the other. Add weights to the left? Add the same to the right. The goal is always to isolate the unknown (get x alone on one side). In engineering, equations describe the balance of forces — a bridge won't stand if the equation is unbalanced.

Definition

Equation

A mathematical statement that two expressions are equal, written with an '=' sign. Solving means finding the value(s) of the variable that make the statement true.

e.g. 3x+5=14⇒x=3\text{e.g. } 3x + 5 = 14 \quad \Rightarrow \quad x = 3

Property / Rule

Additive Inverse Property

Add the same value to both sides of an equation to eliminate a term. This keeps the balance.

If x−5=12, then x−5+5=12+5⇒x=17\text{If } x - 5 = 12, \text{ then } x - 5 + 5 = 12 + 5 \Rightarrow x = 17

Property / Rule

Multiplicative Inverse Property

Multiply or divide both sides by the same non-zero value to isolate the variable.

If 4x=20, then 4x4=204⇒x=5\text{If } 4x = 20, \text{ then } \frac{4x}{4} = \frac{20}{4} \Rightarrow x = 5

Step-by-Step Method for Solving Linear Equations

  1. Step 1: Clear brackets using the distributive law.
  2. Step 2: Clear fractions by multiplying through by the LCD.
  3. Step 3: Move all variable terms to one side, constants to the other.
  4. Step 4: Collect like terms on each side.
  5. Step 5: Divide by the coefficient of the variable.
  6. Step 6: Substitute back to CHECK your answer.

💡 Tip

ALWAYS check your answer by substituting back into the original equation. If both sides are equal, you're correct. This catches sign errors that are easy to make.

Property / Rule

Solving Equations by Factorisation (Zero Product Property)

If a product of factors equals zero, then at least one factor must be zero. Steps: (1) Rearrange so one side equals 0. (2) Factorise the non-zero side completely. (3) Set each factor equal to zero. (4) Solve each mini-equation. (5) Check both solutions.

x2−9=0⇒(x+3)(x−3)=0⇒x+3=0   or   x−3=0⇒x=−3   or   x=3x^2 - 9 = 0 \Rightarrow (x+3)(x-3) = 0 \Rightarrow x+3=0 \;\text{ or }\; x-3=0 \Rightarrow x=-3 \;\text{ or }\; x=3
Worked Examples

Worked Example

Solving equations with brackets and fractions

Problem

Solve for xx: \; 2(3x - 1) - (x + 4) = 3x + 5

Worked Example

Equation with fractions

Problem

Solve for xx:  x3−x−12=1: \; \dfrac{x}{3} - \dfrac{x-1}{2} = 1

Worked Example

Equations involving exponent laws

Problem

Solveforx:(a)  2x=32(b)  3x+1=27xSolve for x: \quad (a)\; 2^x = 32 \quad (b)\; 3^{x+1} = 27^x

Worked Example

Solving equations by factorisation

Problem

Solve for xxusingfactorisation:(a)  x2−16=0(b)  x2+x−6=0 using factorisation: \quad (a)\; x^2 - 16 = 0 \quad (b)\; x^2 + x - 6 = 0
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge2 marks
Solve for xx: 5x−7=185x - 7 = 18
2
L1 · Knowledge2 marks
Solve for xx: 3(x+2)=213(x + 2) = 21
3
L2 · Routine Procedures4 marks
Solve for xx: 4(x−3)−2(x+1)=64(x-3) - 2(x+1) = 6
4
L2 · Routine Procedures3 marks
Solve for xx: 2x+13=5\dfrac{2x+1}{3} = 5
5
L3 · Complex Procedures5 marks
Solve for xx: x+24−x−13=2\dfrac{x+2}{4} - \dfrac{x-1}{3} = 2
6
L3 · Complex Procedures5 marks
A rectangle has perimeter 54 cm. The length is 3 cm more than twice the width. Find the dimensions.
7
L4 · Problem Solving4 marks
Solve for xx: 32x−1=273^{2x-1} = 27
8
L4 · Problem Solving5 marks
Thabo is 4 years older than twice his sister Nandi's age. In 3 years, the sum of their ages will be 43. How old is each person now?
Week 5

2.2 Equations from Word Problems

  • Set up and solve equations from written contexts involving ratio, rate, and proportion
  • Solve problems involving consecutive integers
  • Interpret the solution in context
🌍

Real-World Connection

Every word problem is a hidden equation wearing a disguise. A taxi fare problem, a mixing ratios problem, a speed-distance problem — they all become solvable the moment you write 'Let x = …' and translate the words into algebra. This skill is the foundation of ALL engineering, science, and business mathematics.

Translating Words into Algebra

  • 'Is', 'equals', 'is the same as' → =
  • 'More than', 'increased by', 'added to' → +
  • 'Less than', 'decreased by', 'reduced by' → −
  • 'Times', 'product of', 'multiplied by' → ×
  • 'Divided by', 'per', 'ratio' → ÷
  • 'Consecutive integers': n, n+1, n+2, …
  • 'Consecutive even/odd integers': n, n+2, n+4, …

💡 Tip

Strategy for word problems: (1) Read carefully. (2) Write 'Let x = …' (define your variable with UNITS). (3) Express all other unknowns in terms of x. (4) Write the equation. (5) Solve. (6) Check the answer makes sense in the original problem context.

🌍 Real-World Context

Proportion problems: 'If 3 taps fill a pool in 12 hours, how long for 4 taps?' Use inverse proportion: more taps → less time. So 3×12 = 4×t → t = 9 hours. Direct proportion: more distance → more fuel.

Worked Examples

Worked Example

Consecutive integers problem

Problem

The sum of three consecutive odd integers is 57. Find the integers.

Worked Example

Ratio and proportion word problem

Problem

A mixture of fruit juice is made with apple and orange juice in the ratio 3:2. There are 750 mℓ of apple juice. How much orange juice is needed and what is the total volume?

Worked Example

Age problem

Problem

Thandi is 4 years older than her brother Sipho. In 6 years, the sum of their ages will be 46. Find their current ages.
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge2 marks
Translate into an equation (do not solve): 'Five more than three times a number is 26.'
2
L1 · Knowledge2 marks
The sum of two consecutive integers is 47. Write an equation for this situation.
3
L2 · Routine Procedures3 marks
A number is doubled and then decreased by 11. The result is 15. Find the number.
4
L2 · Routine Procedures4 marks
A pencil costs R3 more than an eraser. 4 pencils and 3 erasers together cost R33. Find the cost of each item.
5
L3 · Complex Procedures4 marks
The sum of three consecutive even integers is 90. Find the integers.
6
L3 · Complex Procedures4 marks
Pens cost R5 each and notebooks cost R18 each. Lerato buys twice as many pens as notebooks and pays R84 in total. How many pens and notebooks did she buy?
7
L4 · Problem Solving5 marks
A car travels 180 km at speed vv km/h. A second car travels the same distance but 30 km/h faster and arrives 1 hour earlier. Set up and solve an equation for vv.
8
L4 · Problem Solving5 marks
Lerato is twice as old as Khotseng. Eight years ago, Lerato was three times as old as Khotseng. How old are they now?

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Algebraic Equations Grade 9 Maths CAPS Notes & Examples | MathSciBuddy