Question 7·Easy·Systems of Two Linear Equations in Two Variables
At a school fundraiser, each T-shirt costs $15 and each hat costs $10. The club sold a total of 120 T-shirts and hats for a combined revenue of $1,550. Which of the following systems of equations could be used to determine the number of T-shirts, , and the number of hats, , sold?
For word problems that ask you to choose a system of equations, do not try to solve for the variables unless asked. Instead, quickly define the variables, then write one equation for the total count (items, people, hours, etc.) and a second equation for totals involving money, distance, or another quantity. Attach each price or rate directly to the correct variable, and assign the correct total to each equation (for example, smaller whole numbers usually represent counts of items, and larger numbers with a dollar sign usually represent money). Finally, scan the answer choices and select the one whose two equations exactly match the ones you wrote.
Hints
Identify what the numbers represent
Ask yourself: Which number (120 or 1,550) is counting how many items were sold, and which number is the amount of money?
Build the equation for number of items
Use the fact that the total number of T-shirts and hats together is 120. How do you show that with and in one simple equation?
Build the equation for total money
Use the prices: $15 for each T-shirt and $10 for each hat. How do you combine $15, $10, , and so that the total equals $1,550?
Desmos Guide
Test a chosen answer choice in Desmos
Pick one answer choice you want to test. In Desmos, treat as and as . Rewrite each equation from that choice so it gives in terms of (solve for in terms of ), then type those two equations into Desmos as lines.
Find and interpret the intersection
Look at where the two lines intersect. The -coordinate of the intersection represents a possible number of T-shirts, and the -coordinate represents a possible number of hats according to that system.
Check whether the intersection matches the story
Take the intersection point from Desmos and check it against the problem: do the two numbers add up to 120, and do they give $1,550 when you compute $15 times (T-shirts) + $10 times (hats)? If not, that answer choice does not model the situation correctly.
Step-by-step Explanation
Define the variables
The problem already defines the variables:
- = number of T-shirts sold
- = number of hats sold
We need two equations involving and : one for the total number of items and one for the total money.
Write the equation for total number of items
The club sold a total of 120 T-shirts and hats. That means the sum of T-shirts and hats is 120.
So the first equation is:
This equation is about counts of items, not money.
Write the equation for total revenue
Each T-shirt costs $15 and each hat costs $10, and together they brought in $1,550.
Money from T-shirts: $15 per T-shirt times T-shirts gives .
Money from hats: $10 per hat times hats gives .
Total revenue equation:
Match your equations to the answer choices
We found the system that models the situation:
This matches choice B, so that is the correct answer.