Question 94·Easy·Systems of Two Linear Equations in Two Variables
A café sells cups of coffee for $3 each and muffins for $2 each. On a certain morning, the café sold a total of 17 items (coffees and muffins) for $40. How many cups of coffee were sold?
For word problems that involve selling different items, always start by defining variables for each item, then write one equation for the total number of items and another for the total cost. Use elimination (often by matching the smaller coefficients, like the muffin price here) to quickly reduce the system to a single-variable equation. Solve, then plug back in and check both the item total and the money total to avoid arithmetic mistakes or answers that only satisfy one of the two conditions.
Hints
Translate the story into algebra
Assign a variable for the number of coffees and another for the number of muffins. How can you write equations for the total number of items (17) and the total cost ($40)?
Set up the system
One equation should use the sum of the items, and the other should use the prices ($3 per coffee and $2 per muffin) to represent the total cost.
Use elimination or substitution
Once you have two equations, try to eliminate one variable. For example, make the coefficients of the same in both equations so you can subtract and get an equation with only .
Check your answer
After finding a value for , compute the corresponding and verify that both the total number of items and the total cost match the problem.
Desmos Guide
Enter the equations
In Desmos, type the two equations using for coffees and for muffins:
x + y = 173x + 2y = 40
Find the intersection point
Look at the graph where the two lines intersect, and click or tap that point. Desmos will show its coordinates .
Interpret the intersection
The -coordinate of the intersection represents the number of cups of coffee, and the -coordinate represents the number of muffins. Compare the -value you see to the answer choices to select the correct option.
Step-by-step Explanation
Define variables and write equations
Let be the number of cups of coffee and be the number of muffins.
From the problem:
- Total items: .
- Total cost: each coffee is $3 and each muffin is $2, so .
So we have the system:
Use elimination to remove one variable
To eliminate , first multiply the first equation by so the terms match:
which gives
Now subtract this new equation from the cost equation:
This simplifies to an equation with only :
Solve for the number of coffees and answer the question
Compute the right-hand side of the equation:
So , which means 6 cups of coffee were sold.
Therefore, the correct answer is 6.