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.
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.
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  $1$1  $3$3  $5$5  $7$7 

$y$y  $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: $72=5$7−2=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
5. Remember to check your answers!
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!
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?
Write an equation for $g$g in terms of $f$f.