3 Linear equations

Lesson

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.

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

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

Summary

The coordinates of the point of intersection of two lines is the solution to the equation of both lines simultaneously.

The following graph displays a system of two equations.

Write down the solution to the system in the form $\left(x,y\right)$(`x`,`y`).

Loading Graph...

Given that the point of intersection of two lines will be the solution to simultaneous equations, one way to solve any pair of simultaneous equations is to plot their graphs and find any points of intersection.

Consider the following linear equations:

$y=3x+6$`y`=3`x`+6 and $y=-x+2$`y`=−`x`+2

Plot the two lines representing these equations on the graph below.

Loading Graph...State the values of $x$

`x`and $y$`y`which satisfy both equations.$x$

`x`= $\editable{}$$y$

`y`= $\editable{}$

Of course, we still need to check whether or not our two lines will intersect graphically. In other words, are the lines parallel or not?

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 $x$`x` and $y$`y` that solve the equations simultaneously.

Consider the following linear equations.

$y=-x-3$`y`=−`x`−3 and $y=-1$`y`=−1

Plot the two lines representing these equations on the graph below.

Loading Graph...State the values of $x$

`x`and $y$`y`which satisfy both equations.$x$

`x`= $\editable{}$$y$

`y`= $\editable{}$

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