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All courses Algebra I · A-CED.2 49 of 76
Create equations in two or more variables and graph them with appropriate labels and scales.

Write a two-variable linear equation from a fixed amount and a rate

Problem
Write a \(\text{two}\text{-}\text{variable}\) linear equation for a cost \(C\) of \(\$12\) plus \(\$4~\text{per}~\text{item}\) \(x\).
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Hint

Identify the fixed amount and the amount per item. The cost starts with \(12\) dollars and increases by \(4\) dollars for each item \(x\).

A linear cost model has the form \(total = rate * quantity + fixed fee\).

Solution walkthrough

01

Name the variables and cost parts

\[\begin{aligned} C=\text{total}~\text{cost} \\ x=\text{number}~\text{of}~\text{items} \\ \text{fixed}~\text{amount}=12~\text{dollars} \\ \text{rate}=4~\text{dollars}/\text{item} \end{aligned}\]

C depends on how many items x are purchased. The $12 is paid once, while $4 is repeated for every item.

02

Write the fixed-plus-rate structure

\[\text{total}=\text{fixed}+\text{rate}*\text{count}\]

A linear fixed-fee model adds the starting amount to the constant rate multiplied by the number of items.

03

Substitute the quantities

\[C=12+4x=4x+12\]

Replace total with C, fixed amount with 12, rate with 4, and count with x. Reordering gives conventional slope-intercept form.

04

Check the interpretation

\[\begin{aligned} x=0~->~C=12 \\ x=1~->~C=16 \end{aligned}\]

At zero items, the model retains the $12 fixed amount. One item adds $4, confirming the equation C=4x+12.

+

Another way

  1. Use slope-intercept form C=mx+b with slope m=4 dollars per item and intercept b=12 dollars.

!

Common mistake

Do not swap 12 and 4. Only the per-item rate multiplies x; the fixed $12 remains constant.

Solution walkthrough video