Linear Relations

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.

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.

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.

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.

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 |

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

Complete the table of values below:

Time (minutes) $15$15 $30$30 $45$45 $60$60 Height (cm)

$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$

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

Complete the table of values below:

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

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.

Complete the table of values:

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

Construct tables of values and graphs, using a variety of tools to represent linear relations derived from descriptions of realistic situations