Skill: Represent linear relations using equations

A linear relation can be written as

$$y=mx+b$$

where:

• $x$ is the input or independent variable.
• $y$ is the output or dependent variable.
• $m$ is the constant rate of change, or slope.
• $b$ is the starting value, or the value of $y$ when $x=0$.

Example: A gym charges a $25 sign-up fee and$8 for each month of membership. 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 $8 for every month, so the slope is

$$m=8$$

Step 3: Identify the starting value

Before any months have passed, the customer still pays the $25 sign-up fee. This is the value when$x=0$, so

$$b=25$$

Step 4: Write the equation

Substitute $m=8$ and $b=25$ into $y=mx+b$:

$$\boxed{y=8x+25}$$

This equation means the total cost is $8$ times the number of months, plus the $25 sign-up fee.

For example, after $3$ months:

$$y=8(3)+25=24+25=49$$

The total cost after $3$ months is 49$.