Select the correct answer.

Chartle is installing new carpet in his rectangular living room, which is 5 feet longer than it is wide. He has received quotes for the total installation costs from two different contractors for the same style of carpet.

Contractor A charges an installation fee of $125 plus $2 per square foot for the carpet.
Contractor B charges an installation fee of $75 plus $2.50 per square foot for the carpet.

Based on this information, which system of equations could be used to determine the width of the living room, $x$, in feet, at which the quote for the installation costs, $y$, is the same for both companies?

A. [tex]
\begin{array}{l}
y=2 x^2+10 x+250 \\
y=2.5 x^2+12.5 x+150
\end{array}
[/tex]

B. [tex]
\begin{array}{l}
y=2 x^2+10 x+125 \\
y=2.5 x^2+12.5 x+75
\end{array}
[/tex]

C. [tex]
\begin{array}{l}
y=2 x^2+130 \\
y=2.5 x^2+80
\end{array}
[/tex]

D. [tex]
y=2 x^2+5 x+125 \\
y=2.5 x^2+5 x+75
[/tex]

Define x as the width and express the length as x + 5 , then calculate the area as x 2 + 5 x .
Formulate the cost equation for Contractor A: y = 2 x 2 + 10 x + 125 .
Formulate the cost equation for Contractor B: y = 2.5 x 2 + 12.5 x + 75 .
The system of equations is: { y = 2 x 2 + 10 x + 125 y = 2.5 x 2 + 12.5 x + 75 ​ .

Explanation

Understanding the Problem Let's break down this problem step by step. We're trying to find a system of equations that represents the cost of carpet installation from two different contractors. The key is to express the total cost, y , in terms of the width of the room, x .

Defining Variables and Area First, let's define our variables:



Let x be the width of the living room in feet.
Since the living room is 5 feet longer than it is wide, the length is x + 5 feet.
The area of the living room is the product of its width and length: A re a = x ( x + 5 ) = x 2 + 5 x This area is crucial because the cost per square foot depends on it.


Contractor A's Cost Now, let's find the total cost for Contractor A. They charge a $ 125 installation fee plus $ 2 per square foot. So, the total cost y can be expressed as: y = 125 + 2 ( x 2 + 5 x ) Simplifying this, we get: y = 125 + 2 x 2 + 10 x y = 2 x 2 + 10 x + 125

Contractor B's Cost Next, let's find the total cost for Contractor B. They charge a $ 75 installation fee plus $ 2.50 per square foot. So, the total cost y can be expressed as: y = 75 + 2.5 ( x 2 + 5 x ) Simplifying this, we get: y = 75 + 2.5 x 2 + 12.5 x y = 2.5 x 2 + 12.5 x + 75

The System of Equations So, the system of equations that represents the total cost for both contractors is:


{ y = 2 x 2 + 10 x + 125 y = 2.5 x 2 + 12.5 x + 75 ​
This system can be used to find the width x at which the total cost y is the same for both contractors.

Final Answer Therefore, the correct system of equations is:

{ y = 2 x 2 + 10 x + 125 y = 2.5 x 2 + 12.5 x + 75 ​
Examples
Imagine you're planning to tile your kitchen floor. You get quotes from two companies: one charges a flat fee plus a rate per square foot, and the other has a different flat fee and rate. Setting up a system of equations like this helps you determine the kitchen size where both quotes would cost the same. This is useful for making cost-effective decisions in home improvement projects, comparing service plans, or even deciding between different investment options where costs and returns vary.

Answered by GinnyAnswer | 2025-07-05