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.

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) |
$1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|

Number of matches ($y$y) |
$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.

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

$x$x |
$1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|

$y$y |
$\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |

**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`=3`x`−5 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 |
$1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|

$y$y |
$-2$−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 |
$1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|

$y$y |
$-2$−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 |
$1$1 | $3$3 | $5$5 | $9$9 |
---|---|---|---|---|

$y$y |
$-2$−2 | $4$4 | $10$10 | $22$22 |

Complete the table for the figures in the given pattern.

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

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

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

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.

Complete the table of values:

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

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