Lesson 17 Year 8 Unit 2 · Linear Relationships +85 XP

Introduction to Simultaneous Equations

Apples cost $2, bananas cost $1. Two purchases, two equations, one answer — that is the power of simultaneous equations: solving two unknowns at the same time.

Think First: Apples cost $2, bananas cost $1. I buy 3 apples and 2 bananas for $8. What if I also buy 2 apples and 4 bananas for $8? Can you find two equations with two unknowns?

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What You Will Master

objectives

Know

Simultaneous equations are two equations with two unknowns
The solution satisfies both equations at once
The solution is the intersection point on a graph

Understand

Why the intersection point is the only point satisfying both equations
There are three types of solutions: one, none, infinite

Can Do

Set up simultaneous equations from word problems
Solve by graphing — find the intersection point
Identify the number of solutions from a graph

Words You Need

vocabulary

Simultaneous equationsTwo or more equations with the same variables solved together to find values satisfying all equations at the same time.

SolutionThe pair of values $(x, y)$ that makes both equations true when substituted.

Intersection pointThe point where two lines cross on a graph; the coordinates are the solution.

Consistent systemA system of equations that has at least one solution (one unique or infinitely many).

Inconsistent systemA system with no solution — the lines are parallel and never intersect.

Dependent systemA system where both equations describe the same line, giving infinitely many solutions.

What Are Simultaneous Equations?

+5 XP

When we have one equation with one unknown, like $2x = 6$, we can solve it easily: $x = 3$.

But what if we have two unknowns? One equation like $x + y = 5$ has many solutions: $(1, 4)$, $(2, 3)$, $(0, 5)$, and so on. To find one specific answer, we need a second equation.

Definition: Simultaneous equations are two (or more) equations that use the same variables and must be solved together. The solution is the pair of values that makes both equations true.

Example system: $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$

Check if $(3, 2)$ is the solution: Equation 1: $3 + 2 = 5$ ✓ Equation 2: $3 - 2 = 1$ ✓. Both satisfied!

A single equation with two unknowns has infinitely many solutions. A second equation pins down the one unique point that works for both.

Intersection point = the solution to both equations

The Graphical Method

+5 XP

To solve simultaneous equations graphically: draw both lines on the same axes, then find where they intersect. The coordinates of that point are the solution.

Why it works: Every point on a line satisfies that line's equation. The intersection is on both lines, so it satisfies both equations.

Watch Me Solve It · Graphical method

Solve graphically: $y = 2x + 1$ and $y = -x + 4$

+15 XP per step

PROBLEM

Solve $\begin{cases} y = 2x + 1 \\ y = -x + 4 \end{cases}$ by graphing.

1

Identify both equations and their features

Equation 1: $y = 2x + 1$ — gradient = 2, $y$-intercept = 1
Equation 2: $y = -x + 4$ — gradient = −1, $y$-intercept = 4
2

Sketch both lines on the same axes
3

Locate the intersection and verify

Lines appear to cross at $x = 1$, $y = 3$. Check: Eq 1: $3 = 2(1) + 1 = 3$ ✓ Eq 2: $3 = -(1) + 4 = 3$ ✓

Solution: $(1, 3)$

Answer$(1, 3)$

Three Types of Solutions

+5 XP

Two lines can relate in three ways:

One unique solution

Different gradients ⇒ lines cross at one point. Example: $y = 2x - 1$ and $y = -\frac{1}{2}x + 3$.

No solution

Same gradient, different intercepts ⇒ parallel lines, never meet. Example: $y = 2x + 1$ and $y = 2x - 2$.

Infinite solutions

Same gradient and same intercept ⇒ same line. Example: $y = x + 1$ and $2y = 2x + 2$.

Pitfall: “No solution” does NOT mean the solution is zero! It means there is no pair of values that satisfies both equations — the lines are parallel.

Remember: Always check if the intersection point actually satisfies both equations. Read $x$ first, then $y$ when reading from a graph.

No solution — Parallel lines:

Setting Up from Word Problems

+5 XP

The first step in solving word problems is to identify the two unknowns and find two different relationships between them.

Strategy:

Define variables (say what $x$ and $y$ represent).
Use the first piece of information to write Equation 1.
Use the second piece of information to write Equation 2.
Solve the system.

Watch Me Solve It · Setting up equations

Café problem: set up the equations

+15 XP per step

PROBLEM

At a café, 2 coffees and 3 muffins cost $13. Three coffees and 1 muffin cost $11. Set up the simultaneous equations.

1

Define the variables

Let $c$ = cost of one coffee (dollars) Let $m$ = cost of one muffin (dollars)
2

Translate the first sentence

“2 coffees and 3 muffins cost $13” → $2c + 3m = 13$
3

Translate the second sentence

“3 coffees and 1 muffin cost $11” → $3c + m = 11$

System: $\begin{cases} 2c + 3m = 13 \\ 3c + m = 11 \end{cases}$

Pro TipAlways write what your variables represent. It earns method marks!

Brain Trainer · 10 quick questions

Brain Trainer · Simultaneous Equations Basics

10 problems

1 Two lines intersect at $(4, 5)$. What is $x$ in the solution?

$x = 4$
2 Two lines intersect at $(-1, 3)$. What is $y$ in the solution?

$y = 3$
3 Two lines both have gradient $m = 3$ but different $y$-intercepts. How many solutions?

No solutions (parallel lines).
4 Two equations describe the exact same line. How many solutions?

Infinitely many solutions.
5 Does $(2, 3)$ satisfy $y = 2x - 1$?

Yes! $3 = 2(2) - 1 = 3$ ✓
6 Does $(2, 3)$ satisfy $y = x + 2$?

No! $3 \neq 2 + 2 = 4$ ✗
7 A line has gradient 2 and passes through $(0, 3)$. Write its equation.

$y = 2x + 3$
8 What does “simultaneous” literally mean?

“At the same time” — both equations must be satisfied together.
9 Lines $y = x + 2$ and $y = -x$ intersect where $x + 2 = -x$. Solve for $x$.

$2x = -2 \Rightarrow x = -1$
10 Which type of system is $y = 2x + 1$ and $y = 2x + 1$? (consistent / inconsistent / dependent)

Dependent (same line, infinitely many solutions).

Quick Check · 5 questions

What does “simultaneous” mean in the context of simultaneous equations?

+10 XP

Two lines intersect at the point $(2, 3)$. What is the solution to the simultaneous equations?

+10 XP

Two lines are parallel. How many solutions are there?

+10 XP

From the graph below, what is the solution to the simultaneous equations?

+10 XP

Which pair of equations has no solution?

+10 XP

Show Your Working · 3 questions

Short Answer Questions

marks available

Apply Medium 5 MARKS

SAQ 1. On the same set of axes, sketch the lines $y = x + 1$ and $y = -x + 3$. Find the solution to this system by identifying the intersection point. Show your working.

Answer in your workbook.

Understand Medium 3 MARKS

SAQ 2. Explain, in your own words, why parallel lines have no simultaneous solution. Use the words “gradient” and “intersect” in your answer.

Answer in your workbook.

Create Medium 4 MARKS

SAQ 3. Write a word problem that could be solved using simultaneous equations. Define your variables and set up the two equations. Do not solve the system.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — ‘Simultaneous’ means at the same time.

2. C — $x = 2$ and $y = 3$ (both values together).

3. A — No solutions (parallel lines never meet).

4. B — $(2, 2)$ (intersection 2 units across and 2 up).

5. D — $y = 2x + 1$ and $y = 2x - 3$ (same gradient $m = 2$, different intercepts).

Short Answer Model Answers

SAQ 1 (5 marks): Line 1: gradient 1, $y$-int 1. Line 2: gradient −1, $y$-int 3. Sketch both [1+1]. At intersection: $x + 1 = -x + 3 \Rightarrow 2x = 2 \Rightarrow x = 1$ [1]. Then $y = 1 + 1 = 2$ [1]. Check: $2 = -(1) + 3 = 2$ ✓ [1]. Solution: $(1, 2)$.

SAQ 2 (3 marks): Parallel lines have the same gradient [1] but different $y$-intercepts. Because they have the same steepness, they never get closer to each other and will never intersect [1]. Since the solution to simultaneous equations is the intersection point, and parallel lines don’t intersect, there is no solution [1].

SAQ 3 (4 marks): Any valid word problem [1] with two unknowns clearly defined [1] and two correct equations [2]. Example: A cinema sells adult tickets for $a$ and child tickets for $c$. 4 adults and 2 children pay $72, 3 adults and 5 children pay $78. Equations: $4a + 2c = 72$ and $3a + 5c = 78$.

Lesson Review

Key Takeaways

Simultaneous equations are two equations with the same two variables, solved together.
The solution is the pair $(x, y)$ that makes both equations true.
Graphically, the solution is the intersection point of the two lines.
One solution: Different gradients ⇒ lines cross once.
No solution: Same gradient, different intercepts ⇒ parallel lines.
Infinite solutions: Same gradient, same intercept ⇒ same line.

Stretch Challenge · +25 XP

Finding Lines Through Points

Challenge 1: A line passes through $(1, 3)$ and $(3, 7)$. Another line passes through $(0, 5)$ and $(2, 1)$. Find the equations of both lines, then solve simultaneously to find their intersection point.

Challenge 2: Consider the system $y = kx + 2$ and $y = 3x - 1$. For what value(s) of $k$ will the system have: (a) one solution, (b) no solution, (c) infinitely many solutions? Explain your reasoning.

Reveal solutions

Challenge 1: Line 1: $m = (7-3)/(3-1) = 2$, using $(1,3)$: $3 = 2(1) + b \Rightarrow b = 1$. So $y = 2x + 1$. Line 2: $m = (1-5)/(2-0) = -2$, $b = 5$. So $y = -2x + 5$. Solve: $2x + 1 = -2x + 5 \Rightarrow 4x = 4 \Rightarrow x = 1$, $y = 3$. Intersection: $(1, 3)$.

Challenge 2: (a) One solution when $k \neq 3$. (b) No solution when $k = 3$ (same gradient, intercepts $2 \neq -1$ are different). (c) Infinitely many: never possible since $2 \neq -1$.

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Lesson Complete!

You have mastered Introduction to Simultaneous Equations.

+50 XP

Unit overview · Maths subject page