Solving Simultaneous Equations Revision Notes for VCE SSCE General Mathematics

Solving Simultaneous Equations

Introduction

When working with two linear equations, we often need to find where they meet. This meeting point is called the point of intersection, and finding it is known as solving the equations simultaneously.

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Key Concept: Simultaneous equations are two or more equations that we solve together to find values that satisfy all equations at the same time.

Understanding intersection points

Two straight lines will meet at a single point unless they are parallel (running in exactly the same direction and never meeting). The coordinates of this meeting point give us the solution to both equations.

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When we find the point of intersection, we are finding the values of $x$ and $y$ that make both equations true at the same time.

Method 1: Finding intersections graphically

The most visual way to solve simultaneous equations is to plot both lines on the same set of axes and identify where they cross.

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Worked Example: Graphical Solution

Consider the equations $y = 2x + 5$ and $y = -3x$ .

Solution:

Looking at where the two lines cross, we can read off the coordinates of the intersection point: (-1, 3).

This means:

$x = -1$
$y = 3$

We can verify this works in both equations:

First equation: $y = 2(-1) + 5 = -2 + 5 = 3$ ✓
Second equation: $y = -3(-1) = 3$ ✓

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Practice Problem

The graphs of $y = 2x - 1$ and $y = 4x$ meet at a single point. By examining where the lines cross on a coordinate plane, you would find their point of intersection.

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Exam Tip: When reading intersection points from a graph, always check both the $x$ and $y$ values carefully against the scale on each axis.

Method 2: Using a CAS calculator to find intersections graphically

Modern graphing calculators can plot the equations and find the exact intersection point automatically. This is more accurate than reading from a hand-drawn graph.

General process for graphical calculator method

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General Steps:

Enter both equations into the calculator's graphing function
Plot both graphs on the same screen
Use the calculator's intersection-finding tool
Read off the exact coordinates

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Worked Example: Calculator Solution

Let's solve $y = 2x + 6$ and $y = -2x + 3$ using a graphing calculator.

After entering both functions and using the intersection tool:

Solution: The calculator shows the intersection point is at x = -0.75 and y = 4.5.

Written as coordinates: (-0.75, 4.5)

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Different calculator brands have different menu systems, but the underlying principle is the same - graph both equations and use the built-in intersection finder.

Method 3: Solving algebraically with a CAS calculator

Instead of graphing the equations, we can use the calculator's algebraic solving functions. This is particularly useful for equations that aren't in $y =$ form.

When to use algebraic methods

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Algebraic solving is especially helpful when:

Equations are written in general form like $24x + 12y = 36$
You need an exact answer rather than reading from a graph
The intersection point has awkward coordinates

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Worked Example: Solving a System Algebraically

Solve the simultaneous equations:

$24x + 12y = 36$

$45x + 30y = 90$

Using a CAS calculator's simultaneous equation solver:

Steps:

Access the simultaneous equation solving function on your calculator
Specify that you have 2 equations with variables $x$ and $y$
Enter the first equation: $24x + 12y = 36$
Enter the second equation: $45x + 30y = 90$
Execute the solve command

Solution: x = 0, y = 3

Verification:

We should always check our solution works in both original equations:

First equation: $24(0) + 12(3) = 0 + 36 = 36$ ✓
Second equation: $45(0) + 30(3) = 0 + 90 = 90$ ✓

Choosing the best method

Method	Best when...	Advantages	Disadvantages
Hand-drawn graph	Visual understanding needed	Easy to see the relationship	Less accurate, time-consuming
Calculator graph	Quick visual solution needed	More accurate than hand-drawn	Still requires reading coordinates
Algebraic (calculator)	Exact answer required	Most accurate, works with any form	Less visual understanding

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Exam Tip: For exam questions, use the method that's most appropriate for the marks available. Simple intersections might be shown graphically, but exact answers typically require algebraic methods.

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Key Points to Remember:

Simultaneous equations are solved together to find values that satisfy all equations at once
The intersection point of two lines gives the solution to both equations
Graphical methods involve plotting both lines and finding where they meet
Algebraic methods use calculation to find exact values without drawing
Parallel lines never intersect, so they have no solution
Always verify your solution by substituting back into both original equations
CAS calculators can solve simultaneous equations both graphically and algebraically

Solving Simultaneous Equations (VCE SSCE General Mathematics): Revision Notes