Question 138·Hard·Linear Equations in Two Variables
A smoothie bar sells two sizes of drinks. Each medium smoothie contains 240 calories, and each large smoothie contains 375 calories. On a certain morning, the bar sold medium smoothies and large smoothies, for a combined total of 6,660 calories. The equation
models this situation.
Which is the best interpretation of in this context?
For questions that ask you to interpret a term in a linear model, first match each symbol to its meaning in the story: what each variable counts and what each number measures (like calories per smoothie). Then look at the structure of the term: a lone variable usually represents a count, a lone number usually represents a per-item amount, and a product like coefficient × variable usually represents a total for that group. Finally, eliminate answer choices that mention the wrong size, the wrong unit (smoothies vs. calories), or include more items than the term actually covers.
Hints
Focus on the meaning of y
Look back at the problem statement. What does the variable stand for in this context?
Think about the number 375
The problem gives you specific calorie amounts. Which size of smoothie has 375 calories?
Interpret a product like 375y
If 375 is the calories per large smoothie and is how many large smoothies were sold, what kind of quantity do you get when you multiply them together?
Desmos Guide
Enter the expressions
Type 240x + 375y on one line in Desmos and 375y on a separate line. When prompted, create sliders for both and .
Change y and watch 375y
Move the slider for to different whole-number values (like 1, 2, 3, 5). Notice how the value of 375y changes: it always equals 375 times the current value of , and its units are calories, not number of smoothies.
Relate 375y to the context
Keep in mind that is the number of large smoothies, and 375 is the calories in each large smoothie. Use what you see in Desmos about 375y (it gives a calorie amount depending only on ) to choose the option that accurately describes this quantity.
Step-by-step Explanation
Identify what the variables represent
From the problem:
- = number of medium smoothies sold
- = number of large smoothies sold
So any expression with in it is describing something about the large smoothies.
Connect the numbers to the story
The problem tells you:
- Each medium smoothie has 240 calories.
- Each large smoothie has 375 calories.
So:
- = calories in each medium smoothie
- = calories in each large smoothie
When a per-item amount (like calories per smoothie) is multiplied by a number of items (like number of smoothies), the result is a total amount for that group.
Interpret the expression 375y
In :
- is the calories in each large smoothie.
- is the number of large smoothies sold.
- Their product is the total calories coming from the large smoothies.
Therefore, represents the total number of calories in all large smoothies sold.