A point on the $xy$xy-plane represents a pair of quantities, the $x$x-value and the $y$y-value. We can write this pair in the form $\left(x,y\right)$(x,y), which we call an ordered pair. We say that a set of points on the $xy$xy-plane forms a linear relationship if we can pass a single straight line that goes through all the points.
Each column in a table of values may be grouped together in the form $\left(x,y\right)$(x,y), which we know as an ordered pair. Let's consider the following table of values:
$x$x | $1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|
$y$y | $-2$−2 | $1$1 | $4$4 | $7$7 |
The table of values has the following ordered pairs:
$\left(1,-2\right),\left(2,1\right),\left(3,4\right),\left(4,7\right)$(1,−2),(2,1),(3,4),(4,7)
We can plot each ordered pair as a point on the $xy$xy-plane.
Points plotted from the table of values |
We can plot the ordered pair $\left(a,b\right)$(a,b) by first identifying where $x=a$x=a along the $x$x-axis and $y=b$y=b along the $y$y-axis.
Take $\left(3,4\right)$(3,4) as an example. We first identify $x=3$x=3 along the $x$x-axis and draw a vertical line through this point. Then we identify $y=4$y=4 along the $y$y-axis and draw a horizontal line through that point. Finally we plot a point where the two lines meet, and this represents the ordered pair $\left(3,4\right)$(3,4).
In the example above, we can draw a straight line that passes through these points like so:
Straight line passing through all four points |
So we say that these points form a linear relationship.
We may also be interested in finding out whether an additional point also lies on the straight line that passes through the other points. Consider the ordered pair $\left(0,-5\right)$(0,−5). If we plot this ordered pair on the $xy$xy-plane, then we obtain the following:
Straight line passing through additional point |
So, we can say that the additional point lies on the line or that all five points form a linear relationship.
Alternatively, we can refer to the table of values to determine whether the ordered pair $\left(0,-5\right)$(0,−5) lies on the line. Let's refer back to the table of values from before, but consider the additional point when $x=0$x=0:
$x$x | $0$0 | $1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|---|
$y$y | $\editable{}$ | $-2$−2 | $1$1 | $4$4 | $7$7 |
The value of $y$y decreases by $3$3 as the value of $x$x decreases by $1$1. So we expect to find $y=-5$y=−5 when $x=0$x=0. This is exactly the ordered pair $\left(0,-5\right)$(0,−5), which tells us that this point satisfies the linear relationship between $x$x and $y$y.
Do the points on the plane form a linear relationship?
Yes
No
Consider the table of values given below.
$x$x | $-2$−2 | $-1$−1 | $0$0 | $1$1 |
---|---|---|---|---|
$y$y | $6$6 | $2$2 | $-2$−2 | $-6$−6 |
Plot the points in the table of values.
Is the relationship in the table of values linear?
No
Yes
Consider the table of values given below.
$x$x | $-2$−2 | $-1$−1 | $0$0 | $1$1 |
---|---|---|---|---|
$y$y | $2$2 | $-1$−1 | $-4$−4 | $-7$−7 |
Does the ordered pair $\left(-4,8\right)$(−4,8) satisfy the linear relationship between $x$x and $y$y?
No
Yes