Question 55·Easy·Linear Equations in Two Variables
Last month, a freelance coder billed clients for hours of front-end work and hours of back-end work. The equation
represents the total amount, in dollars, the coder earned. What is the best interpretation of the number in this context?
For interpretation-of-equations questions, do not try to solve the equation. Instead, first identify what each variable represents and what units it has. Then look at how the numbers are attached to those variables: coefficients usually represent rates (like dollars per hour), and constant terms represent fixed amounts. Match the coefficient with the correct variable (front-end vs. back-end in this problem) and check whether the choice describes a rate or a total, eliminating any option that mixes up the type of work or confuses hours with dollars per hour.
Hints
Look at what x and y stand for
Check the sentence that introduces and . What does measure, and what does measure?
Think about what 40x and 55y represent
If and are hours of work, what units should and have so that they can add up to a total amount of money?
Connect the coefficient to the type of work
The number 55 is attached to . Is front-end or back-end work, and does a coefficient usually represent hours worked or dollars per hour?
Desmos Guide
Enter the expression to see the structure
Type 40x + 55y into a Desmos expression line. Notice that the numbers 40 and 55 are directly in front of the variables and ; these are the coefficients.
Connect coefficients to their meanings
Recall from the problem that and are measured in hours of front-end and back-end work. Ask: what units must the coefficients (40 and 55) have so that, when multiplied by hours, they give dollars? Then focus specifically on the coefficient of and interpret it as a rate for the type of work represented by .
Step-by-step Explanation
Identify what the variables represent
From the problem, is the number of hours of front-end work and is the number of hours of back-end work the coder did last month.
So:
- : hours of front-end work
- : hours of back-end work
Interpret each term in the equation
The equation is
This represents the total money earned.
- The term means dollars earned for each hour of front-end work, times hours.
- The term means dollars earned for each hour of back-end work, times hours.
So the numbers multiplying the variables (40 and 55) are rates in dollars per hour.
Focus on the meaning of 55
The number 55 is the coefficient of in .
- is measured in hours of back-end work.
- To turn hours into dollars, we multiply by a dollars-per-hour rate.
Therefore, 55 must be the dollar amount earned for each hour of back-end work. In words, the correct interpretation is: The freelance coder earned $55 per hour for back-end work last month.