When an  equation has exactly two variables in it, we can express the relationship between those variables on the number plane.
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.
x | -2 | -1 | 0 | 1 | 2 |
y | 1 | 2 | 3 | 4 | 5 |
We can now plot these as ordered pairs (x,\,y) on the number plane:
\left(-2,\,1\right),\, \left(-1,\,2\right),\, \left(0,\,3\right),\, \left(1,\,4\right),\, \left(2,\,5\right)
Consider the equation y=3x.
Complete the table of values.
x | -5 | -3 | -1 | 1 |
y |
Plot the points in the table of values.
To complete a table of values for an equation, substitute each given x-value into the equation to find the corresponding y-value.
To plot the points from a table, move horizontally along the x-axis according to the x-value, and move vertically along the y-axis according to the corresponding y-value for each pair of values in the table.
When plotting a set of points onto a number 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 number plane that passes through all the plotted points, otherwise the relationship is non-linear.
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 number 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, where m and c are numeric values.
For example: y=-3x+4 is linear with m=-3 and c=4, but y=x^{2}-1 is not linear because it includes the term x^{2}.
Consider the equation y=x+5. A table of values is given below.
x | 1 | 2 | 3 | 4 |
y | 6 | 7 | 8 | 9 |
Plot the points in the table of values.
Do the points on the plane form a linear relationship?
A relationship is linear if we can draw a straight line on the number plane that passes through all the plotted points, otherwise the relationship is non-linear.
Any linear relationship can be represented by an equation of the form y=mx+c, where m and c are numeric values.
So far in this lesson we have looked at equations that have exactly two variables, x and y. However, we can also plot points for equations that have only one variable.
Consider the equation y=4.
Notice that the value of x does not affect the value for y, since x is not even in the equation. No matter what value x takes, the equation is only true if the value for y is 4.
x | -2 | -1 | 0 | 1 | 2 |
y | 4 | 4 | 4 | 4 | 4 |
What about the equation x=-2?
x | -2 | -2 | -2 | -2 | -2 |
y | 0 | 1 | 2 | 3 | 4 |
When we talk about the number plane, we usually think of the xy-plane. However, the axes of the number plane are not limited to representing just x and y-values.
Unless specified otherwise, the left number in an ordered pair will correspond to the variable on the horizontal axis.
A dead tree is 7 metres tall but since the tree is dead, its height does not change over time.
Let the height of the tree be h metres and the time passed be y years.
Which three of the following ordered pairs, in the form \left(y,\,h\right), match the growth of the dead tree?
Which of the following graphs shows the growth of the dead tree?
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 number plane, it will be a horizontal or vertical line.
Unless specified otherwise, the left number in an ordered pair will correspond to the variable on the horizontal axis.