Points on a Line II

As we previously saw, a linear equation describes a line which we can draw from a table of values. Each point that we plot onto the $xy$xy-plane represents just one of the infinitely many points that make up the line. Is there a way to know if a given point, say $\left(a,b\right)$(a,b), lies on the line without having to graph the line?

Let's look at an example of substituting a point into an equation.

Worked example

Does $\left(4,-36\right)$(4,−36) lie on the line $y=-8x-4$y=−8x−4?

Think: Checking to see whether a point lies on a line requires us to substitute the point $\left(4,-36\right)$(4,−36) into the equation $y=-8x-4$y=−8x−4 and seeing if both sides of the equation are equal.

Do: Substituting the point $\left(4,-36\right)$(4,−36) means to replace $x$x with $4$4 and $y$y with $-36$−36 into the equation $y=-8x-4$y=−8x−4.

Let's first substitute $x=4$x=4 into the right hand side and see if it equals our expected value of $y$y:

Of course, replacing $y$y with $-36$−36 simply gives us $-36$−36 and we can see that both sides of the equation are equal at the point $\left(4,-36\right)$(4,−36). So the point lies on the line.

The strategy of substituting the coordinate of a point to determine whether it lies on the line may be used in reverse to find values of an unknown variable. Consider the following example.

Worked example

The equation of a line is $3x-y-4=0$3x−y−4=0 and it passes through point $\left(3,k\right)$(3,k).

Find the value of $k$k.

Think: We know that we can substitute the point $\left(3,k\right)$(3,k) into the equation because we're told the point lies on the line.

Do: To substitute, we replace $x$x with $3$3 and $y$y with $k$k in the equation $3x-y-4=0$3x−y−4=0.

Now the last equation is the solution, and we can swap each side to write it as $k=5$k=5.

Practice Examples

Question 1

Question 2

QUESTION 3