Question 25·Easy·Systems of Two Linear Equations in Two Variables
A coffee shop sells two sizes of brewed coffee. A small coffee costs $3, and a large coffee costs $5. On a certain Tuesday, the shop sold a total of 180 coffees and collected $720 in coffee sales.
Which of the following systems of equations can be used to find the number of small coffees, , and large coffees, , the shop sold that day?
For “write the system” word problems, start by clearly labeling your variables, then write one equation for each distinct relationship in the story (often one for a total count and one for money or another measure). Make sure the units match on both sides of each equation: totals in items should equal item counts (like 180 coffees), and totals in dollars should equal money amounts (like 720). Finally, scan the choices and quickly eliminate any equation where coefficients don’t match the given prices or where a ‘coffees’ equation is set equal to a dollar amount, or vice versa.
Hints
Think about the total number of items
Ignore the prices for a moment. How can you express “a total of 180 coffees” using for small coffees and for large coffees?
Think about the total amount of money
Each small coffee is $3 and each large is $5. How can you write an expression for the total money from small coffees and the total from large coffees, and what should their sum equal?
Check units (coffees vs. dollars)
In each equation, make sure both sides measure the same thing. One equation should compare numbers of coffees to 180, and the other should compare dollars to 720.
Desmos Guide
Represent variables in Desmos
In Desmos, use for the number of small coffees and for the number of large coffees (so corresponds to and to ).
Test each answer choice as a system
For each option (A, B, C, D), type its two equations into Desmos, replacing with and with . For example, for an equation like , type x + y = 180.
Find and interpret the intersection point
For each system, look at the intersection point of the two lines. The coordinates give a pair of values for the numbers of small and large coffees. Check whether these values are positive whole numbers, add up to 180, and when plugged into the cost expression give a total of 720 dollars. The system whose intersection point meets all these conditions matches the situation.
Step-by-step Explanation
Identify what the variables represent
The problem defines the variables for us:
- = number of small coffees sold
- = number of large coffees sold
We must write equations that connect and to (1) the total number of coffees and (2) the total amount of money.
Write the equation for total number of coffees
The shop sold a total of 180 coffees, and every coffee is either small or large.
So, the number of small coffees plus the number of large coffees must equal 180:
This equation uses the same units (coffees) on both sides.
Write the equation for total money collected
Each small coffee costs $3, and each large coffee costs $5.
- Money from small coffees is dollars.
- Money from large coffees is dollars.
Together, they collected $720 in sales, so the sum of these amounts must equal 720:
This equation uses dollars on both sides.
Match the system to the answer choices
From the earlier steps, the system that models the situation is
Looking at the answer choices, this system appears in choice D, so D is correct.