Question 3·Hard·Linear Equations in Two Variables
To earn money for college, Avery works two part-time jobs, A and B. She earns $10 per hour at job A and $20 per hour at job B. In one week, Avery earned a total of $s working at the two jobs. The graph shows all possible combinations of the numbers of hours she could have worked at the two jobs to earn $s.
What is the value of ?
For this type of SAT question, first translate the situation into a linear equation: total earnings = (rate at job A)×(hours at A) + (rate at job B)×(hours at B), so . Then use a simple point from the graph—usually an intercept where one job’s hours are 0—because that makes one term drop out and lets you compute with a single multiplication. Avoid multiplying axis labels by each other; always multiply hours by the correct hourly rate.
Hints
Turn the situation into an equation
Let the horizontal axis represent hours at job A and the vertical axis represent hours at job B. How can you write an expression for Avery’s total pay in terms of those two numbers of hours?
Focus on an intercept
Look at where the line crosses one of the axes (for example, the horizontal axis). What does that point tell you about how many hours Avery worked at each job?
Use a point to find the total pay
For an intercept where she works only one of the jobs, how would you calculate her total pay using that job’s hourly rate and the number of hours from the graph?
Desmos Guide
Use Desmos as a calculator with a point from the graph
From the graph, pick any clear point on the line (for example, an intercept where one of the coordinates is 0). Suppose that point corresponds to hours at job A and hours at job B. In Desmos, type the expression 10*x + 20*y using the actual numbers from that point (e.g., 10*16 + 20*0). The value Desmos outputs is the total earnings .
Step-by-step Explanation
Write an equation for Avery’s total earnings
Let be the number of hours Avery works at job A and be the number of hours she works at job B.
- At job A, she earns $10 per hour, so her earnings from A are .
- At job B, she earns $20 per hour, so her earnings from B are .
So her total weekly earnings are given by the equation
The graph shows all pairs that satisfy this equation for one fixed value of .
Use a convenient point from the graph
Look at the intercepts of the line on the graph.
From the graph, the line crosses the horizontal axis (job A hours) at when .
That point means: Avery worked 16 hours at job A and 0 hours at job B, but still earned the same total amount as in any other point on the line.
Calculate the total earnings for that point
If Avery works 16 hours at job A and 0 hours at job B, her total earnings are
Compute this:
- So
Therefore, , so the correct answer is 160.