When an equation has exactly two variables in it, we can express the relationship between those variables on the Cartesian plane.
Equations, tables and graphs
We can make a table of values for any equation by substituting some test values for one of the variables and recording the values that the other variable must be for the equation to be true. Doing this can help us see the relationship between the two variables.
Let's start with the equation $y=x+3$y=x+3. By substituting a range of values for $x$x into the equation, we can find the corresponding $y$y-values to make this table:
| $x$x | $-2$−2 | $-1$−1 | $0$0 | $1$1 | $2$2 |
|---|---|---|---|---|---|
| $y$y | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 |
We can now plot these as ordered pairs $\left(x,y\right)$(x,y) on the Cartesian plane:
$\left(-2,1\right),\left(-1,2\right),\left(0,3\right),\left(1,4\right),\left(2,5\right)$(−2,1),(−1,2),(0,3),(1,4),(2,5)
We can start by plotting the point $\left(-2,1\right)$(−2,1). Since this point has a negative $x$x-value and a positive $y$y-value, we know that it will be in the second quadrant of the Cartesian plane. Knowing this, we can plot the point $\left(-2,1\right)$(−2,1) by starting at the origin and moving $2$2 units to the left, then $1$1 unit up.
Another way that we can plot this ordered pair on the Cartesian plane is by finding the point of intersection of the lines through $x=-2$x=−2 and $y=1$y=1:
After plotting all the ordered pairs onto the Cartesian plane, we form a line:
Practice question
Question 1
Consider the equation $y=3x$y=3x.
Fill in the blanks to complete the table of values.
$x$x $-5$−5 $-3$−3 $-1$−1 $1$1 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Plot the points in the table of values.
Loading Graph...
Linear relationships
When plotting a set of points onto a Cartesian plane, the relationship between the points will be linear if they all lie on a straight line, and non-linear otherwise.
A relationship is linear if we can draw a straight line on the Cartesian plane that passes through all the plotted points, otherwise the relationship is non-linear.
Worked Example
Do the points on the plane form a linear relationship?
Think: Can we draw a straight line on the graph that passes through all the plotted points?
Do: If this is a linear relationship then any two points should lie on the straight line. Knowing this, we should be able to draw our straight line through any two of the plotted points.
In this case, we can choose our two end points:
Since the straight line passes through all the plotted points, the points on the plane form a linear relationship.
Reflect: When checking if a set of points forms a linear relationship, we can choose any two of the points and draw a straight line through them. If the points form a linear relationship then any two points will result in a straight line passing through all the points.
Exploration
Let's have a look at some equations and their graphs.
 |
For each of the equations, we have filled in a table of values and plotted the ordered pairs onto the Cartesian plane. On each graph, we have also drawn a straight line through two of the points to check if the relationship is linear or not.
As we can see from the graphs above, (a) and (c) represent linear relationships while (b) represents a non-linear relationship.
Is there a connection between an equation and whether or not the relationship between its variables is linear?
Any linear relationship can be represented by an equation of the form $y=mx+c$y=mx+c, where $m$m and $c$c are numeric values.
For example: $y=-3x+4$y=−3x+4 is linear with $m=-3$m=−3 and $c=4$c=4, but $y=x^2-1$y=x2−1 is not linear because it includes the term $x^2$x2.
Practice question
Question 2
Consider the equation $y=x+5$y=x+5. A table of values is given below.
| $x$x | $1$1 | $2$2 | $3$3 | $4$4 |
|---|---|---|---|---|
| $y$y | $6$6 | $7$7 | $8$8 | $9$9 |
Plot the points in the table of values.
Loading Graph...Do the points on the plane form a linear relationship?
Yes
ANo
B
Constant relationships
So far in this lesson we have looked at equations that have exactly two variables, $x$x and $y$y. However, we can also plot points for equations that have only one variable.
Exploration
Consider the equation $y=4$y=4.
Notice that the value of $x$x does not affect the value for $y$y, since $x$x is not even in the equation. No matter what value $x$x takes, the equation is only true if the value for $y$y is $4$4. So our table of values will look like this:
| $x$x | $-2$−2 | $-1$−1 | $0$0 | $1$1 | $2$2 |
|---|---|---|---|---|---|
| $y$y | $4$4 | $4$4 | $4$4 | $4$4 | $4$4 |
If we plot these points onto the Cartesian plane we can see that this relationship is linear.
As we can see, the line $y=4$y=4 is a horizontal line through the marker for $4$4 on the y-axis.
What about the equation $x=-2$x=−2?
Using the same logic, we can get a table of values:
| $x$x | $-2$−2 | $-2$−2 | $-2$−2 | $-2$−2 | $-2$−2 |
|---|---|---|---|---|---|
| $y$y | $0$0 | $1$1 | $2$2 | $3$3 | $4$4 |
and a graph:
As we can see, this relationship is also linear, except in this case the line $x=-2$x=−2 is vertical.
A constant relationship is a kind of linear relationship where the value of the variable does not change. When we express this relationship using an equation, we only use one variable with no powers. When we draw this relationship on the Cartesian plane, it will be a horizontal or vertical line.
Naming the axes
When we talk about the Cartesian plane, we usually think of the $xy$xy-plane. However, the axes of the Cartesian plane are not limited to representing just $x$x and $y$y-values.
Suppose we wanted to plot the relationship between time and distance, with$t$t representing time and $d$d representing distance. To do this, we would need to change the axis names from $x$x and $y$y to be time and distance.
Notice that we also include $d$d next to distance and $t$t next to time. This is because those are the variables that are used to represent them.
For these axes, any ordered pair should be in the form $\left(t,d\right)$(t,d).
Unless specified otherwise, the left number in an ordered pair will correspond to the variable on the horizontal axis.
Practice question
Question 3
A dead tree is $7$7 metres tall but since the tree is dead, its height does not change over time.
Let the height of the tree be $h$h metres and the time passed be $y$y years.
Which three of the following ordered pairs, in the form $\left(y,h\right)$(y,h), match the growth of the dead tree?
$\left(3,6\right)$(3,6)
A$\left(0,7\right)$(0,7)
B$\left(7,21\right)$(7,21)
C$\left(10,1\right)$(10,1)
D$\left(4,7\right)$(4,7)
E$\left(2,7\right)$(2,7)
FWhich of the following graphs shows the growth of the dead tree?
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...D