Question 136·Easy·Systems of Two Linear Equations in Two Variables
A school fundraiser sold T-shirts for $10 each and hats for $7.50 each. The fundraiser sold a total of 200 items and collected $1,700. Which of the following systems of equations could be used to determine the number of T-shirts, , and hats, , sold?
For “write the system” questions, always start by labeling the variables clearly, then turn each key sentence of the word problem into its own equation: a count equation (just variables and a total number of items) and a value equation (price × quantity summed to equal a total dollar amount). Match each part of the story carefully to an equation, paying special attention to which numbers are counts vs. money and which coefficients go with which variables, and then pick the choice whose two equations exactly match what you wrote.
Hints
Use the definitions of t and h
is the number of T‑shirts and is the number of hats. What equation should connect and to the total number of items, 200?
Separate “number of items” from “amount of money”
One equation should involve just the counts of T‑shirts and hats, and the other should involve the prices ($10 and $7.50) and the total money, $1,700.
Build the money equation carefully
How do you express the money from T‑shirts in terms of ? How do you express the money from hats in terms of ? Add those two expressions and set them equal to $1,700.
Desmos Guide
Translate the story into equations first
Write down your own two equations from the problem: one for the total number of items (just and ) and one for the total money (using 10 and 7.5 with and and setting it equal to 1700). Do this on paper before using Desmos.
Enter your system into Desmos
In Desmos, type your first equation on one line (for example, something like t + h = 200 using x and y instead of and ), and your second equation on the next line (your money equation). Desmos will graph the two lines and show their intersection point, which represents the solution.
Compare with each answer choice
For each answer choice, replace your equations in Desmos with the two equations from that choice (again using x and y). Check whether those equations match the structure of the equations you wrote from the story: one should be a simple item-count equation, and the other should be a money equation with 10 on the T‑shirt variable and 7.5 on the hat variable, totaling 1700.
Step-by-step Explanation
Identify what the variables represent
The problem defines the variables for you:
- = number of T‑shirts sold
- = number of hats sold
Every equation in the system must be built using these variables and the information given: total items and total money.
Write an equation for the total number of items
The fundraiser sold a total of 200 items, and the items are just T‑shirts and hats.
- T‑shirts:
- Hats:
Therefore, the quantity equation (counting items) must be:
Write an equation for the total amount of money
Each T‑shirt costs $10, and each hat costs $7.50. The total money collected is $1,700.
- Money from T‑shirts: $10 each, so
- Money from hats: $7.50 each, so
Adding these amounts gives the money equation:
Match your equations to the answer choices
We now have the system that correctly models the situation:
- Total items:
- Total money:
The answer choice that shows exactly these two equations together is the correct one.