Points on a Line I

Lesson

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.

Exploration

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 $y$y $1$1 $2$2 $3$3 $4$4 $-2$−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 $y$y $\editable{}$ $1$1 $2$2 $3$3 $4$4 $-2$−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.

Practice Questions

Question 1

Do the points on the plane form a linear relationship?

1. Yes

A

No

B

Yes

A

No

B

question 2

Consider the table of values given below.

 $x$x $y$y $-2$−2 $-1$−1 $0$0 $1$1 $6$6 $2$2 $-2$−2 $-6$−6
1. Plot the points in the table of values.

2. Is the relationship in the table of values linear?

No

A

Yes

B

No

A

Yes

B

question 3

Consider the table of values given below.

 $x$x $y$y $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $-1$−1 $-4$−4 $-7$−7
1. Does the ordered pair $\left(-4,8\right)$(4,8) satisfy the linear relationship between $x$x and $y$y?

No

A

Yes

B

No

A

Yes

B