A bookstore sold a total of 90 books in one day. Each fiction book sold for $12, and each nonfiction book sold for $18. Let be the number of fiction books sold and let be the number of nonfiction books sold. The total number of books and the total sales in dollars can be modeled by the system of equations
According to this model, how many nonfiction books were sold?
A system of linear equations uses two conditions to pin down the same two unknowns. If the equations already model a story, type them on separate lines in Desmos and click where the lines cross. Interpret that point in the problem’s variable order: reporting the first coordinate when the question asks for the second is a trap.
Hints
- Hint 1
A system means both equations must hold for the same pair of numbers. Graph the book-count equation first. What other condition will narrow its many possible pairs to one?
- Hint 2
An intersection is a point where two graphs cross, so it satisfies both equations. Enter the sales equation on a second line and click the crossing rather than estimating its coordinates.
- Hint 3
An ordered pair lists first and second. Since counts nonfiction books, the second coordinate of the intersection is the one you need.
Step-by-step
Approach 1: Read the Desmos intersection
Step 1Graph the book-count condition
Type in Desmos. Every point on this line represents fiction and nonfiction counts that add to . But one line cannot fix both counts; you also need the sales condition.
- Step 2
Read the nonfiction count
Add on a second line and click where the lines cross. Desmos shows . The intersection satisfies both totals. Because the point is written , the second coordinate is the nonfiction count. So nonfiction books were sold. Grid in 70.
Approach 2: Compare sales with an all-fiction day
Step 1Find the all-fiction sales amount
Imagine all books were fiction at $12 each. Type in Desmos; it shows . That is the all-fiction sales amount, before accounting for any higher-priced nonfiction books.
- Step 2
Find the extra sales per nonfiction book
Each nonfiction book sold for $18 instead of $12. Subtract the fiction price from the nonfiction price:
So each nonfiction book adds $6 above the all-fiction amount.
- Step 3
Divide the extra sales by the extra per book
The store made dollars more than the all-fiction amount. Divide that difference by $6 per nonfiction book: type in Desmos. It shows , so nonfiction books were sold. Grid in 70.