In the previous lesson we looked at some simultaneous equations and verified solutions by substituting in the values of coordinate pairs. In this lesson, we will look at a method of solving using the graphs of the equations.
Linear equations can be represented as straight line graphs. To solve a pair of linear simultaneous equations, we find the values that satisfy both equations, and graphically this means finding the point of intersection of the two straight lines.
For example, let’s find the solution to the simultaneous equations:
$y=x+1$y=x+1 and $y=-2x+7$y=−2x+7
We want to plot the two equations as graphs. Remember, there are two common ways to visualise linear equations as graphs:
- through finding their $x$x- and $y$y-intercepts or
- by rearranging into gradient-intercept form.
Here are the graphs of the two straight lines:
We can then see that there is only one point of intersection, and it is at $\left(2,3\right)$(2,3). So the solution that satisfies the two equations is $x=2$x=2 and $y=3$y=3.
Parallel lines
In Section 1.01, we saw that not all pairs of simultaneous linear equations have a solution. If we think about this graphically, is it possible to graph two straight lines that never intersect?
Of course, it happens when they're parallel! We can tell for sure that two lines are parallel when they have the same gradient. The best way to find the gradient of a line is to rearrange the linear equation into gradient-intercept form, $y=mx+b$y=mx+b, and $m$m will be our gradient.
For example, the linear equations $y=3x-1$y=3x−1 and $y=3x+6$y=3x+6 both have a gradient of $3$3, so they are parallel lines. The parallel lines don’t meet, so the pair of simultaneous linear equations has no solution.
Use the following applet to practice graphing two linear equations to find the solution of the system if it exists.
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Practice questions
Question 1
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).
Question 2
Consider the following linear equations.
$y=x-1$y=x−1 and $y=-2x+8$y=−2x+8
Find the coordinates of the intercepts of the line $y=x-1$y=x−1.
$x$x-intercept $\left(\editable{},\editable{}\right)$(,) $y$y-intercept $\left(\editable{},\editable{}\right)$(,) Find the coordinates of the intercepts of the line $y=-2x+8$y=−2x+8.
$x$x-intercept $\left(\editable{},\editable{}\right)$(,) $y$y-intercept $\left(\editable{},\editable{}\right)$(,) Plot the lines of the two equations on the same graph.
Loading Graph...State the coordinate pair $\left(x,y\right)$(x,y) which satisfies both equations.
$\left(x,y\right)=\left(\editable{},\editable{}\right)$(x,y)=(,)
Question 3
Consider the following equations:
$4x-2y=2$4x−2y=2
$-2x+4y=2$−2x+4y=2
Plot the two lines.
Loading Graph...State the pair of $x$x and $y$y values which satisfy both equations.
$x$x$=$=$\editable{}$
$y$y$=$=$\editable{}$