# Table of values

Lesson

Given two variables, $x$x and $y$y, is there a way to show how these two variables are related? At the very least, we may be able to see certain values of $y$y that occur at certain values of $x$x. We can collect this information into a table of values.

#### Exploration

Imagine we started with a triangle made out of matchsticks. We can form a pattern by adding two additional matchsticks each time as shown below.

The table of values for this pattern connects the number of triangles made ($x$x) with the number of matches needed ($y$y).

 Number of triangles ($x$x) Number of matches ($y$y) $1$1 $2$2 $3$3 $4$4 $3$3 $5$5 $7$7 $9$9
Table of values

A table of values is a table used to present the quantities of two variables that are related in some way.

As we saw before, a table of values may be used to describe a pattern. However, we may also be given an equation or a rule to describe the relationship between two variables. Let's take a look below.

#### Worked example

Consider the equation $y=3x-5$y=3x5. Using this rule, we want to complete the following table of values.

 $x$x $y$y $1$1 $2$2 $3$3 $4$4 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$

Think: We wish to find the value of $y$y at each value of $x$x in the table of values.

Do: First we find the value of $y$y when $x=1$x=1 by substitution.

Substituting $x=1$x=1 into $y=3x-5$y=3x5 we end up with:

$y=3\times\left(1\right)-5$y=3×(1)5

Which simplifies to give:

$y=-2$y=2

So after finding the value of $y$y when $x=1$x=1, we have:

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

Reflect: In general, we can complete a table of values by repeating this process of substitution for each variable given in the table.

Completing the rest of the table of values gives us:

 $x$x $y$y $1$1 $2$2 $3$3 $4$4 $-2$−2 $1$1 $4$4 $7$7

For a table of values, the values of $x$x do not need to increase by one each time. We could obtain the following table of values repeating the same procedure as before:

 $x$x $y$y $1$1 $3$3 $5$5 $9$9 $-2$−2 $4$4 $10$10 $22$22

#### Practice Examples

##### Question 1

The height of a candle is measured every $15$15minutes.

1. Complete the table of values below:

 Time (minutes) Height (cm) $15$15 $30$30 $45$45 $60$60 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$

##### Question 2

Consider the equation $y=5x+6$y=5x+6.

1. Complete the table of values below:

 $x$x $y$y $-10$−10 $-5$−5 $0$0 $5$5 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$

##### QUESTION 3

A racing car starts the race with $140$140 litres of fuel. From there, it uses fuel at a rate of $2$2 litres per minute.

1. Complete the table of values:

 Number of minutes passed ($x$x) Amount of fuel left ($y$y) $0$0 $5$5 $10$10 $15$15 $20$20 $70$70 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$