Question 132·Hard·Linear Equations in Two Variables
Yesterday, a clothing store sold a total of 72 T-shirts, some of which were red and the rest blue. Each red T-shirt cost $12 and each blue T-shirt cost $18. If the total revenue from the T-shirt sales yesterday was $1,080, how many red T-shirts did the store sell?
(Express the answer as an integer)
For mixture or sales problems with totals, immediately define variables for each type of item and write two equations: one for the total quantity and one for the total cost. Turn it into a system, then use substitution or elimination to reduce it to a one-variable linear equation. Keep your algebra organized (distribute carefully and combine like terms step by step), and at the end, check that your solution makes sense with both the total number and total revenue, and that you’re giving the quantity the question actually asks for (red vs. blue, adult vs. child, etc.).
Hints
Define variables clearly
Let one variable represent the number of red T‑shirts and another variable represent the number of blue T‑shirts. What two facts does the problem give you that you can turn into equations?
Write a system of equations
Use the total number of shirts to write an equation like "red + blue = total." Then use the prices and total revenue to write a second equation involving the same variables.
Reduce to one variable
From the easier equation (the one with just the total number of shirts), solve for one variable in terms of the other, and substitute into the revenue equation so you only have one variable to solve for.
Finish solving and interpret
After simplifying, you will get a one-step equation such as . Solve for and remember that the question is asking for the number of red T‑shirts, not blue.
Desmos Guide
Enter equations for the system
In the first line, type y = 72 - x (this represents with as the number of red shirts and as the number of blue shirts).
Enter the revenue equation in terms of y
In the second line, type 12x + 18y = 1080. Desmos will prompt you to convert this to a function; select y = and it will automatically solve for so you can graph it.
Find the intersection point
Look for the point where the two lines intersect. The -coordinate of this intersection gives the number of red T‑shirts, and the -coordinate gives the number of blue T‑shirts.
Step-by-step Explanation
Define variables and write equations
Let be the number of red T‑shirts and be the number of blue T‑shirts.
From the problem:
- Total shirts: .
- Total revenue: .
Now we have a system of two equations in two variables.
Use substitution to get an equation in one variable
From the first equation, solve for in terms of :
Substitute this into the revenue equation:
becomes
Now simplify:
- Distribute 18: .
- Combine like terms: .
- Subtract 1296 from both sides: .
So you now have a simple one-variable equation: .
Solve for the number of red T‑shirts
Solve by dividing both sides by :
So the store sold 36 red T‑shirts.