 # 3.07 Solving simultaneous equations graphically

Lesson

## Introduction

When we want to solve two equations at the same time, we are looking for the values our variables can take such that both equations will be true. This is referred to as solving simultaneous equations.

## Equations as graphs

The equations that we will be working with are used to describe the relationship between two variables, usually x and y. As such, we can express these equations graphically as lines on the xy-plane.

If we have two lines on the xy-plane, any point at which those two lines intersect will have an x and y-value satisfying both equations.

The coordinates of the point of intersection of two lines is the solution to the equation of both lines simultaneously. So one way to solve any pair of simultaneous equations is to plot their graphs and find any points of intersection.

If two lines are parallel and are not the same line, they will never intersect. Otherwise, there will be some point of intersection that we can use to solve the equations simultaneously.

Since parallel lines always have the same slope, we can use the gradient of two equations to check whether or not their graphs will have a point of intersection. If their gradients are the same then there will be no values for x and y that solve the equations simultaneously.

### Examples

#### Example 1

Consider the following linear equations:

\begin{aligned} y&=5x-7 \\ y &=-x+5 \end{aligned}

a

Sketch the two lines representing these equations on a cartesian plane.

Worked Solution
Create a strategy

Use the gradient and y-intercept from the equations to graph the lines.

Apply the idea

For the line y=5x-7, m=5=\dfrac{5}{1} and c=-7. So we can plot the y-intercept then move to the right by the run of 1.4 and down by the rise of -7 to find another point on the line.

Now we can draw the line through these two points.

For the line y=-x+5, m=-1=-\dfrac{1}{1} and c=5. So we can plot the y-intercept then move to the right by the run of 5 and down by the rise of 0 to find another point on the line. Then we can draw the line through these two points.

From the graph, we can see that the point of intersection is (2,\,3).

b

State the values of x and y which satisfy both equations.

Worked Solution
Create a strategy

Use the point of intersection of the two lines.

Apply the idea

From the point of intersection (2,\,3), the values that satisfy both equations are x=2 and y=3.

Idea summary

We can solve any pair of simultaneous equations by plotting their graphs and finding the point of intersection.

If two different lines have the same gradient, then they will be parallel and will have no point of intersection. So their equations will have no simultaneous solution.

### Outcomes

#### VCMNA337

Solve simultaneous linear equations, using algebraic and graphical techniques including using digital technology