Linear Equations

New Zealand

Level 6 - NCEA Level 1

Lesson

We've already learnt about linear equations, which showed a relationship between two variables. Now we are going to look at a special kind of linear relationship called a proportional relationship.

Two quantities are said to be proportional if *they vary in such a way that one is a constant multiple of the other*. In other words, they always vary by the same constant. We can also calculate the unit rate in a proportional relationship, which tells us how much the dependent variable will changes with a one unit increase in the independent variable.

For example, if the cost of some items is always five times the number of items, we can say that this is a proportional relationship because there is a constant multiple between the cost and the number of items - $5$5. We can write these proportional relationships as linear equations. The example above could be written as $y=5x$`y`=5`x` and again we can see that the coefficient of $x$`x` describes the constant of the proportional relationship.

We will learn more about the constant of proportionality and writing proportional relationships as equations later but now let's focus on determining whether relationships are proportional or not.

Remember!

A relationship is proportional if there is a constant multiple between the two variables.

We can also compare proportional relationships to make judgements about rates of change.

Consider the equation $y=7x$`y`=7`x`.

What is the gradient of $y=7x$

`y`=7`x`?Select the graph that below that shows $y=7x$

`y`=7`x`.Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...D

Irene and Valentina are both making handmade birthday cards. Irene can make $8$8 cards every $13$13 minutes. Valentina can make $6$6 cards every $15$15 minutes.

Plot this information on the graph.

Loading Graph...How can you tell who was quicker at making cards?

The steeper line relates to the faster maker

AThe shallower line relates to the faster maker

B

Oliver is making cups of fruit smoothie. The amount of bananas and strawberries he uses is shown in the proportion table.

Strawberries | $4$4 | $8$8 | $12$12 | $16$16 | $20$20 |
---|---|---|---|---|---|

Bananas | $5.5$5.5 | $11$11 | $16.5$16.5 | $22$22 | $27.5$27.5 |

Graph this proportional relationship.

Loading Graph...What is the unit rate of this relationship?

Select ALL the statements that describe the proportional relationship.

For every $5.5$5.5 bananas Oliver uses, he adds $4$4 strawberries.

AThe unit rate of bananas in respect to strawberries is $\frac{11}{8}$118.

BFor every $4$4 bananas, Oliver uses $5.5$5.5 strawberries.

CThe unit rate of bananas in respect to strawberries is$\frac{8}{11}$811.

D

We'll learn later about direct proportional relationships and inverse proportional relationships.