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11 Linear relationships
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Stage 4

11.06 The point of intersection

Lesson
Worksheet
Practice
Lesson

Introduction

We have looked at how to solve different types of  equations  algebraically by applying reverse operations. Now we are going to look at how to solve equations graphically.

Solve by graphs

A linear graph will extend forever in both directions. This means that two unique lines must intersect at some point, unless the two lines have the same gradient and are therefore parallel. We can use this fact to solve equations graphically by plotting two straight lines on a number plane and finding the coordinates of the point where the two lines cross. This is known as the point of intersection.

We can use this method to solve simple equations like 4x=8 and 3+x=8, and also much harder equations, containing  variables on both sides  such as 2x-7=-8x+13. Let's explore how we can solve these three equations graphically.

Examples

Example 1

In this question we will find the point of intersection between the line y=x-9 and the line y=-x-7.

a

Sketch the line y=x-9, together with the line y=-x-7.

Worked Solution
Create a strategy

Use the y-intercept and another point to sketch the lines.

Apply the idea
-2
-1
1
2
3
x
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
y

For y=x-9, the y-intercept is (0,-9). Sincem=1. we can find the next point on the line by moving 1 unit right then 1 unit up to get to (1,-8).

Now we can draw the line through these two points.

For y=-x-7, the y-intercept is (0,-7). Since m=-1, we can find the next point on the line by moving 1 unit right then 1 unit down to get to (1,-8).

Now we can draw the line through these two points.

b

What is the point of intersection of these two lines?

Worked Solution
Create a strategy

Look for the point where the lines meet.

Apply the idea

The point of intersection is at (1,-8).

Reflect and check

This means that x=1 is the solution to the equation x-9=-x-7. We can verify this algebraically:

\displaystyle x-9\displaystyle =\displaystyle -x-7Write the equation
\displaystyle x-9+x\displaystyle =\displaystyle -7Add x to both sides
\displaystyle 2x-9\displaystyle =\displaystyle -7Simplify
\displaystyle 2x\displaystyle =\displaystyle -7+9Add 9 to both sides
\displaystyle 2x\displaystyle =\displaystyle 2Simplify
\displaystyle x\displaystyle =\displaystyle 1Divide both sides by 2
Idea summary

We can find the solution to an equation graphically by finding the x-coordinate of the point of intersection of two lines.

Outcomes

MA4-11NA

creates and displays number patterns; graphs and analyses linear relationships; and performs transformations on the Cartesian plane

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