Simultaneous Equations

Master systems of linear equations

CAPS Grade 10 Mathematics

This topic forms part of the CAPS-aligned Grade 10 Mathematics curriculum and focuses on solving systems of two linear equations with two variables. Each section includes interactive games and quizzes to test your understanding.

Learning Outcomes

  • Understand the concept of simultaneous equations
  • Solve systems using substitution and elimination methods
  • Interpret solutions graphically as intersection points
  • Formulate equations from word problems
  • Interpret solutions in real-world contexts

Introduction to Simultaneous Equations

Simultaneous equations involve two or more equations with two or more variables. The solution must satisfy all equations simultaneously, representing the point where their graphs intersect.

General Form
a₁x + b₁y = c₁
a₂x + b₂y = c₂
Key Concept: A solution to simultaneous equations is a pair (x, y) that makes BOTH equations true. Graphically, this is the point where the two lines intersect.

Solving Methods

1

Substitution Method

Solve one equation for one variable, substitute into the other

x + y = 5
2x - y = 1

x = 5 - y
2(5-y) - y = 1
10 - 3y = 1 → y = 3
x = 2 → (2,3)
2

Elimination Method

Add/subtract equations to eliminate one variable

x + y = 5
2x - y = 1

Add: 3x = 6 → x = 2
2 + y = 5 → y = 3
(2,3)
3

Choosing a Method

  • Substitution: When one variable is easy to isolate
  • Elimination: When coefficients are opposites or easy to match

Quiz 1 - Solving Methods

Solve: x + 2y = 7, 3x - y = 4

A) x=1, y=3
B) x=2, y=2.5
C) x=3, y=2
D) x=1, y=2

Word Problems

Number Problems

Problem

The sum of two numbers is 20, and their difference is 4. Find the numbers.

Solution
x + y = 20
x - y = 4
Add: 2x = 24 → x = 12, y = 8

Cost Problems

Problem

3 pens and 2 pencils cost R21. 2 pens and 3 pencils cost R19. Find cost of each.

Solution
3p + 2c = 21
2p + 3c = 19
Multiply first by 3, second by 2:
9p + 6c = 63, 4p + 6c = 38
Subtract: 5p = 25 → p = 5, c = 3

Quiz 2 - Word Problems

Two numbers sum to 15, difference is 5. What are they?

A) 10 and 5
B) 12 and 3
C) 8 and 7
D) 9 and 6

Graphical Interpretation

One Solution

Lines intersect at one point (different slopes)

Example: x+y=5, 2x-y=1
Intersection at (2,3)

No Solution

Parallel lines (same slope, different intercept)

Example: x+y=5, x+y=7

Infinite Solutions

Same line (same slope and intercept)

Example: x+y=5, 2x+2y=10

Quiz 3 - Graphical

How many solutions do parallel lines have?

A) One
B) None
C) Infinite
D) Two

Practice & Assess

Match - Method to Example

x = 5 - y
Substitution
Add equations
Elimination
Intersection point
Graphical
Let x = pens
Word Problem

Fill - Elimination

x + y = 5
2x - y = 1
Adding gives: __x = 6

Practice Questions

Q1

Solve by substitution: x + 2y = 7, 3x - y = 4

Q2

Solve by elimination: 2x + 3y = 11, 4x - y = 3

Q3

Two numbers differ by 5. Their sum is 19. Find them.

Answers

Q1: x=3, y=2 | Q2: x=2, y=3 | Q3: 12 and 7

Summary of Key Concepts

Methods: Substitution, Elimination, Graphical
Word Problem Steps: Read → Define variables → Form equations → Solve → Check
Types of Solutions: One solution (intersecting), No solution (parallel), Infinite solutions (same line)

Key Terms

Simultaneous System Substitution Elimination Graphical Intersection Parallel Solution Set Variable Coefficient
Quadratic Equations Linear Inequalities