Represent linear relations using equations

A linear relation is expressed as an equation in two variables, commonly $y=mx+b$ , where $m$ is the constant rate of change (slope) and $b$ is the value of $y$ when $x=0$ . The equation connects and translates information from a context, table, or graph, including positive, negative, and zero rates of change; nonlinear, piecewise, and more abstract parameterized relations are outside this scope.

Detailed Explanation: 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$.

Learn by doing: Represent linear relations using equations

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