Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Question

The freshman and sophomore classes at Allenville High School are decorating floats for homecoming. The freshmen have already spent $458 on their float, plus they need to buy floral sheeting that costs $46 per roll. The sophomores, who have spent $453 so far on theirs, still need to purchase vinyl grass at $47 per roll. Both classes plan to buy the same number of rolls, since they have the same area to cover. By coincidence, the two floats will have the same total cost in the end. How many rolls will each class be buying? How much will each class spend in total?

By buying
rolls, the two classes will each spend $
in total.

Answer 1

To formulate a system of equations for this problem, we can define the following variables:

• Let \( x \) be the number of rolls each class buys.
• Let \( F \) be the total cost for the freshmen.
• Let \( S \) be the total cost for the sophomores.

From the problem, we know:

1. The freshmen's total cost can be expressed as: \[ F = 458 + 46x \]

2. The sophomores' total cost can be expressed as: \[ S = 453 + 47x \]

Since the total costs for both classes are equal, we can set the two equations equal to each other: \[ 458 + 46x = 453 + 47x \]

Now we will solve this equation for \( x \).

Step 1: Simplify and solve for \( x \)

1. Move all terms involving \( x \) to one side of the equation and constant terms to the other side: \[ 458 - 453 = 47x - 46x \] \[ 5 = x \]

So, the freshmen and sophomores will each buy \( x = 5 \) rolls.

Step 2: Calculate the total cost for each class

Using \( x = 5 \):

• For the freshmen: \[ F = 458 + 46 \times 5 = 458 + 230 = 688 \]

• For the sophomores: \[ S = 453 + 47 \times 5 = 453 + 235 = 688 \]

Both classes will spend the same amount.

Final Answers:

By buying 5 rolls, the two classes will each spend $688 in total.