NZ Level 5
Finding the Rule
Lesson

When we look at linear relationships, we are looking at a relationship between two variables. For example, if we were to say that the total cost of items ($C$C) is three times the number of items ($N$N), we could write this algebraically as $C=3N$C=3N. This algebraic expression is also called a rule.

We often use a table of values to display the values we record. Then we can find the relationship between the two variables and write it as a rule.

Remember!

A linear relationship is in the form $y$y$=$=$a\times x$a×x$+$+$b$b, where $a$a and $b$b are any numbers. $x$x and $y$y are the most common variables used, but we can replace these with other letters to show the relationship between any two variables.

Linear relationships increase or decrease at a constant rate. This means that $y$y (variable 1) changes the same amount for each unit of change in $x$x, (variable 2). This constant rate of increase or decrease is the gradient of the line.

## Working out the rule

We'll run through the process of working out the rule by looking at an example.

Here is a typical table of values:

 $x$x $y$y $1$1 $3$3 $5$5 $7$7 $7$7 $11$11 $15$15 $19$19

1. Figure out by how much both amounts are increasing or decreasing .

We can see that each time the $x$x variable increases by $2$2, $y$y increases by $4$4

2. Adjust variable 2 so that it is increasing or decreasing at the same rate as variable 1

When we adjust $x$x so that it increases at the same rate as $y$y, we get $2x$2x, so we're going to compare $2x$2x and $y$y now. NB: The coefficient of $x$x in our equation will be $2$2.

3. Find the remaining difference after adjusting between variable 1 and variable 2.

When $y=7$y=7, we see that $2x=2$2x=2, and when $y=11$y=11, $2x=6$2x=6. Therefore the difference is always the same. Finding the difference we get: $7-2=5$72=5, so our constant term is $5$5.

4. Form the equation by putting together adjusted rate and the difference

$y=2x+5$y=2x+5

Let's sub another pair of points from our table of values into our equation to make sure its right.

 $\text{LHS }$LHS $=$= $15$15 $\text{RHS }$RHS $=$= $2\times5+5$2×5+5 $=$= $15$15 $=$= $\text{LHS }$LHS

Since our equation is balanced, we know we've got it right!

#### Examples

##### Question 1

Consider the pattern for blue boxes.

a) Complete the table

b) Write a formula that describes the relationship between the number of blue boxes ($b$b) and the number of columns ($c$c).

c) How many blue boxes will there be if this pattern were to continue for $38$38 columns?

d) If this pattern continued and we had $45$45 blue boxes. How many columns would we have?

##### Question 2

Write an equation for $g$g in terms of $f$f.

### Outcomes

#### NA5-9

Relate tables, graphs, and equations to linear and simple quadratic relationships found in number and spatial patterns.