Represent linear relationships using equations

A linear relationship is represented by an equation such as $y=mx+b$ , where $m$ is the constant rate of change and $b$ is the initial value, and these quantities correspond to patterns in tables, graphs, and descriptions. The relationship may be proportional when $b=0$ or nonproportional when $b\ne0$ , with positive, negative, whole-number, and rational rates interpreted in context; slope and intercept are not interchangeable. This scope excludes nonlinear relationships, systems of equations, and more advanced function generalizations.

Detailed Explanation: 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.

Learn by doing: Represent linear relationships using equations

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