# 6.04 Completing tables of values

Lesson

Given two variables, say $x$x and $y$y, that are connected in some way, we can describe the relationship between the two variable using a formula, a graph or a table of values.

A table of values shows certain values of $y$y that occur for certain values of $x$x - this gives us a snapshot of the relationship. This can often be the first step in finding a formula or finding pairs to graph the relationship.

#### 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.

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 and asked to create a table that shows a snapshot of the relationship for certain values. 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. You may also be able to complete the table by noticing a pattern.

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 questions

##### Question 1

Complete the table for the figures in the given pattern.

1.  Step number ($x$x) Number of matches ($y$y) $1$1 $2$2 $3$3 $4$4 $5$5 $10$10 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$

##### Question 2

Complete the table of values using the formula $q=2p-3$q=2p3.

1.  $p$p $q$q $0$0 $1$1 $2$2 $3$3 $4$4 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$

##### QUESTION 3

There are $20$20 litres of water in a rainwater tank. It rains for a period of $24$24 hours and during this time the tank fills up at a rate of $8$8 litres per hour.

1. Complete the table of values:

 Number of hours passed ($x$x) Amount of water in tank ($y$y) $0$0 $4$4 $6$6 $7$7 $9$9 $11$11 $12$12 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$

### Outcomes

#### ACMEM036

substitute given values for the other pronumerals in a mathematical formula to find the value of the subject of the formula