Linear Algebra

Master linear equations and inequalities

CAPS Grade 10 Mathematics

This topic forms part of the CAPS-aligned Grade 10 Mathematics curriculum and focuses on solving linear equations and inequalities, forming the foundation for advanced algebra. Each section includes interactive games and quizzes to test your understanding.

Learning Outcomes

  • Define and identify linear equations and inequalities
  • Solve linear equations using algebraic techniques
  • Solve linear inequalities and represent solutions on number lines
  • Translate word problems into algebraic equations/inequalities
  • Apply problem-solving strategies to real-world contexts
  • Verify solutions and check for reasonableness
1

Introduction to Linear Equations

1.1 Definition and Representation

A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable.

S

Standard Form

ax + b = 0

where a and b are constants, a ≠ 0, x is the variable

F

Function Form

y = mx + c

where m is slope, c is y-intercept, x and y are variables

1.2 Solving Linear Equations

+/-

Addition/Subtraction Property

Add or subtract same quantity from both sides

If a = b, then a ± c = b ± c
×/÷

Multiplication/Division Property

Multiply or divide both sides by same non-zero quantity

If a = b, then ac = bc (c ≠ 0)

Solving Basic Linear Equation

Example

Solve: 2x + 5 = 11

Solution
  • Subtract 5: 2x = 6
  • Divide by 2: x = 3
  • Verify: 2(3) + 5 = 11

Quiz 1 - Linear Equations

Solve: 3x - 7 = 8

A) x = 3
B) x = 5
C) x = 7
D) x = 15
2

Introduction to Linear Inequalities

2.1 Definition and Representation

A linear inequality compares two expressions using inequality symbols:

<
>
=
=

2.2 Solving Linear Inequalities

Critical Rule: When multiplying or dividing both sides by a negative number, you MUST reverse the inequality sign.

Basic Inequality

Example

Solve: 3x - 2 < 7

Add 2: 3x < 9
Divide by 3: x < 3

Inequality with Negative Coefficient

Example

Solve: -2x > 6

Divide by -2 (REVERSE sign): x < -3

2.3 Representing Solutions on Number Line

○

Open Circle

For strict inequalities: < or >

x = 3

x < 3

•

Closed Circle

For inclusive inequalities: ≤ or ≥

x = -2

x ≥ -2

Quiz 2 - Linear Inequalities

Solve: -3x ≤ 12

A) x ≥ -4
B) x = -4
C) x = 4
D) x = 4
3

Applications and Word Problems

3.2 Problem-Solving Strategy

1

Understand the Problem

Read carefully. Identify knowns, unknowns, and relationships.

2

Develop a Plan

Translate into algebraic equation/inequality.

3

Carry Out the Plan

Solve using appropriate techniques.

4

Look Back

Check if solution makes sense in original context.

Word Problem Example

Problem

"The sum of a number and 5 is less than 12. Find the possible values of the number."

Solution
  • Let x = the number
  • x + 5 < 12
  • x < 7
  • The number can be any value less than 7.

Quiz 3 - Word Problems

"A number decreased by 8 is at least 15." What inequality represents this

A) x - 8 > 15
B) x - 8 = 15
C) 8 - x > 15
D) 8 - x = 15

Practice & Assess

Match - Inequality Symbols

<
Less than
>
Greater than
=
Less than or equal
=
Greater than or equal

Fill - Solve for x

4x - 5 = 11
x = ___

Common Mistakes to Avoid

Mistake 1

Not reversing inequality sign when dividing by negative:

Wrong: -3x > 9 → x > -3

Correct: -3x > 9 → x < -3

Mistake 2

Distributive property errors:

Wrong: 2(x + 3) = 2x + 3

Correct: 2(x + 3) = 2x + 6

Mistake 3

Forgetting to verify solutions

Always substitute your solution back into the original equation.

Practice Exercises

Basic Equations

  1. 5x + 3 = 18
  2. 2(x - 4) = 10
  3. (x/3) + 2 = 5

Inequalities

  1. 3x + 2 = 11
  2. -4x < 12
  3. 2x - 5 = 3x + 1

Word Problems

  1. Find three consecutive integers whose sum is 48.
  2. Rectangle: length is 3 more than twice width, perimeter 36cm.
  3. Adult R50, Child R30. Total tickets 200, revenue at least R8000.
Answers

Q1: x=3 | Q2: x=9 | Q3: x=9 | Q4: x=3 | Q5: x>-3 | Q6: x=-6

Key Terms

Linear Equation Linear Inequality Variable Constant Coefficient Slope y-intercept Number Line Open Circle Closed Circle Solution Set Verify

Assessment Preparation

Classwork & Homework

  • Practice daily exercises
  • Show all steps clearly
  • Check answers systematically

Tests & Exams

  • Read questions carefully
  • Allocate time wisely
  • Show working for partial credit

Projects & Investigations

  • Apply concepts to real situations
  • Use clear mathematical language
  • Present solutions logically
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