Question 105·Hard·Systems of Two Linear Equations in Two Variables
At a school fundraiser, T-shirts and mugs were sold at fixed prices.
- On Monday, 4 T-shirts and 3 mugs sold for a total of $95.
- On Tuesday, the price of a T-shirt was reduced by 20%, but the price of a mug stayed the same. That day, 3 T-shirts and 5 mugs sold for a total of $105.
What was the price, in dollars, of one mug before the discount?
For problems about prices on different days, immediately assign variables to the unknown prices and translate each day’s description into an equation. Convert percent changes into multipliers (a 20% discount becomes times the original price) to avoid confusion. This typically gives a system of two linear equations in two variables; use elimination to remove one variable and solve quickly. When fractions appear, clear them by multiplying the whole equation, and always check that your solution satisfies all given totals to catch setup or arithmetic errors before you move on.
Hints
Set up variables and Monday's equation
Let one variable represent the original T-shirt price and another represent the mug price. Use the information from Monday to write an equation involving 4 T-shirts, 3 mugs, and a total of $95.
Turn the 20% discount into a multiplier
A 20% discount means the customer pays 80% of the original price. Write the new T-shirt price in terms of the original price using (or ), then use Tuesday’s totals to form a second equation.
Solve the system efficiently
You now have two equations with the same two variables. Try using elimination: adjust one or both equations so the coefficient of matches, then subtract to solve for the mug price .
Check your result against both days
Once you get a value for the mug price, plug it back into both the Monday and Tuesday equations to see if the totals match $95 and $105. If either total is off, there was a mistake in your setup or algebra.
Desmos Guide
Enter the variables and Monday equation
In Desmos, let represent the original T-shirt price and represent the mug price. Type the first equation as 4x + 3y = 95.
Enter the Tuesday equation with the discount
Use for the discounted T-shirt price on Tuesday. Type the second equation as 3(0.8x) + 5y = 105 (Desmos will graph this line as well).
Find the intersection point
Look for the point where the two lines intersect. Tap or click that intersection; Desmos will show the coordinates . The -coordinate of this point is the mug price. Compare that value to the answer choices and select the matching option.
Step-by-step Explanation
Define variables and use Monday's information
Let be the original price of one T-shirt (in dollars), and let be the price of one mug (in dollars).
From Monday:
- 4 T-shirts cost .
- 3 mugs cost .
- Together they cost $1.
So the Monday equation is:
Interpret the 20% discount and use Tuesday's information
A 20% discount means the T-shirt now costs 80% of its original price. In decimal form, 80% is , so the new T-shirt price is .
On Tuesday:
- 3 T-shirts cost .
- 5 mugs cost .
- Together they cost $1.
So the Tuesday equation is:
Simplify :
To avoid decimals, rewrite as :
which simplifies to
Multiply the whole equation by 5 to clear the fraction:
Use elimination to solve the system
Now you have this system of equations:
To eliminate , make the -coefficients match. Multiply the first equation by 3:
which gives
Now subtract this new equation from the second equation:
Solve for the mug price and choose the answer
From the last equation,
Divide both sides by 16:
So the price of one mug before the discount was $15. The correct answer choice is C) 15.