Question 10·Easy·Linear Equations in Two Variables
Nut Order Costs
The College Coffee Shop orders almonds and cashews for its baked goods. Almonds cost $4 per pound and cashews cost $6 per pound. The total cost , in dollars, for ordering pounds of almonds and pounds of cashews is modeled by the equation:
Which of the following is the best interpretation of 25 in this context?
For linear-model interpretation questions, first identify what each variable represents in the context, then match each coefficient and term to a real-world meaning: numbers multiplied by variables usually represent per-unit rates (like dollars per pound), and constant terms (numbers not multiplied by variables) usually represent fixed amounts that do not change with the variables. Check what the expression gives when the variables are zero to understand the constant term, then carefully match this meaning to the most precise answer choice.
Hints
Look at which terms change with the order size
Compare the parts of the equation that include or to the part that does not. Which parts will get bigger if you order more pounds of nuts, and which part will stay the same?
Think about what happens if you order nothing
Imagine and . What is then, and what would that number mean in the real situation?
Match the meaning to the choices
Once you know what 25 represents in the situation, look for the choice that describes a cost that does not depend on how many pounds of nuts are ordered.
Desmos Guide
Enter the cost formula
Type C = 4a + 6c + 25 into Desmos. Desmos will treat this as a relationship between , , and ; you can also create a table with columns a, c, and an expression for C such as 4a + 6c + 25.
Test different order sizes to see what 25 does
In a table, try different values for and (for example, ) and watch how changes. Notice that each time you increase by 1, increases by 4, and each time you increase by 1, increases by 6, but there is always the same extra 25 dollars added to the nut costs—this shows the role of the constant term.
Step-by-step Explanation
Recognize the structure of the cost equation
The equation is
Here:
- = pounds of almonds
- = pounds of cashews
- = total cost in dollars
The numbers 4 and 6 are each multiplied by a number of pounds, but 25 is added on by itself.
Interpret the terms with variables
The term means 4 dollars times each pound of almonds. So 4 is the price per pound of almonds.
The term means 6 dollars times each pound of cashews. So 6 is the price per pound of cashews.
Both of these change when or change, so they are variable parts of the cost.
Interpret the constant term
The number 25 is not multiplied by or , so it does not change when the number of pounds changes.
If you ordered 0 pounds of almonds and 0 pounds of cashews (so and ), the equation gives
That means you would still pay 25 dollars even with no nuts ordered, so 25 must represent a cost that stays the same for every order, no matter how many pounds you buy.
Match this meaning to the answer choices
A cost that stays the same for every order and is added on top of the weight-based nut costs is best described as a fixed delivery charge added to the order, which matches answer choice C.