Skill: Represent linear relationships using equations

A linear relationship can be written in the form

$$y=mx+b$$

• $m$ is the constant rate of change: how much $y$ changes when $x$ increases by $1$.
• $b$ is the initial value: the value of $y$ when $x=0$.

Example

A gym charges a one-time sign-up fee of 12$and then$$5$ each month. Write an equation for the total cost.

Step 1: Choose the variables

Let

• $x$ = number of months
• $y$ = total cost in dollars

Step 2: Identify the rate of change

The cost increases by 5$ for every month, so the rate is

$$m=5$$

Step 3: Identify the initial value

Before any months have passed, the sign-up fee is already 12$. Thus, when$x=0$,

$$b=12$$

Step 4: Write the equation

Substitute $m=5$ and $b=12$ into $y=mx+b$:

$$\boxed{y=5x+12}$$

This equation means that the total cost is 5$per month plus the initial$$12$ fee.

For example, after $3$ months:

$$y=5(3)+12=15+12=27$$

The total cost is 27$. Because the initial value is not$0$, this is a nonproportional linear relationship.