Question 113·Medium·Systems of Two Linear Equations in Two Variables
A sporting goods store sells a home-team jersey and an away-team jersey. Together, the original prices total $160. The store applies a 15% discount to the home-team jersey and a 30% discount to the away-team jersey, and then a coupon takes an additional $10 off the combined discounted price. The final amount paid for both jerseys is $107. Which system of equations gives the original price , in dollars, of the home-team jersey, and the original price , in dollars, of the away-team jersey?
For systems word problems, first assign clear variables and translate each sentence into an equation, one idea at a time. For percent discounts, remember that a discount of means you pay of the original price, so use a decimal like for . Then handle any fixed coupons or fees by adding or subtracting a constant term in the equation. Finally, match your equations to the answer choices, paying close attention to whether the constant is added or subtracted and whether coefficients represent what you pay, not what you save.
Hints
Start with the original total
Focus on the sentence that says the original prices together total $160. How can you write that as an equation using and ?
Turn percent discounts into what you actually pay
If an item is discounted by , what percent of the original price do you still pay? Do the same for a discount, then write each as a decimal times the original price.
Represent the coupon correctly
After applying the percent discounts, the coupon takes $10 off the combined price. Does that mean you should add or subtract 10 in your equation? Where should the 107 go?
Desmos Guide
Enter each answer choice as a system
For each answer choice, replace with and with and type the two equations into Desmos, for example: x + y = 160 on one line and the corresponding second equation from that choice on another line. Do this for options A, B, C, and D (you can hide and unhide them using the checkboxes).
Find and interpret the intersection point
For each system, look at where the two lines intersect; that point represents the jersey prices that satisfy that choice's equations. Think of as the home-jersey price and as the away-jersey price.
Check the story using a separate expression
In a new Desmos line, type the expression 0.85x + 0.70y - 10. Then, for each answer choice, click its intersection point and see what value this expression takes at that point. The system that correctly models the situation will be the one whose intersection point makes this expression equal to the stated final cost of 107, with both prices positive and reasonable.
Step-by-step Explanation
Define the variables and translate the total original price
Let be the original price of the home-team jersey and be the original price of the away-team jersey, both in dollars.
The problem says: "Together, the original prices total $160." That means the sum of the two original prices is $160, which gives the equation
Model the percent discounts on each jersey
The home-team jersey has a discount. Paying less means you pay of its original price. In decimal form, , so the amount paid for the home jersey is .
The away-team jersey has a discount. Paying less means you pay of its original price. In decimal form, , so the amount paid for the away jersey is .
So after the percent discounts, the total (before the coupon) is
Incorporate the $10 coupon into the equation
The coupon "takes an additional $10 off the combined discounted price." That means you subtract $10 from the total after the percent discounts.
So the amount actually paid is
The problem tells you that this final amount equals 107, so you set this expression equal to 107:
Write the full system of equations
Putting both relationships together, you get a system of two equations in and :
- Original prices total $160:
- Final paid amount after discounts and coupon is 107:
So the correct system is