Skill: Represent linear relations using tables

A table represents a linear relation by matching each input with its corresponding output.

• The independent quantity is the input. You choose its value.
• The dependent quantity is the output. Its value depends on the input.
• In a linear relation, equal increases in the input produce equal increases in the output.

Example: A gym charges a one-time fee of $5 plus $3 for each class attended. Make a table showing the total cost for different numbers of classes.

Let

• $x$ = number of classes, the independent quantity
• $y$ = total cost in dollars, the dependent quantity

The cost rule is

$$y=5+3x$$

The $5$ is the initial fee, and $3$ is the cost for each class.

Choose input values for $x$ and calculate the matching output:

Number of classes, $x$	Total cost, $y=5+3x$
$0$	$5+3(0)=5$
$1$	$5+3(1)=8$
$2$	$5+3(2)=11$
$3$	$5+3(3)=14$

So the table is

Each row gives an ordered pair:

$$(0,5),\ (1,8),\ (2,11),\ (3,14)$$

Notice that when the number of classes increases by $1$, the cost always increases by 3$:

$$8-5=3,\qquad 11-8=3,\qquad 14-11=3$$

This constant change shows that the table represents a linear relation. The value $5$ is the initial value because it is the output when $x=0$.