Air Conditioning Installation

Which company is the best choice to do the job?

Which company should you choose to install an Air Conditioning System?

John's family is looking for an Air Conditioning company to install their 4 air conditioners. They do not know which company they should choose to meet their needs. They want it to be installed in every bedroom in the house, but they are not sure how long it will take them. They only know that it will take a maximum time of 10 hours.

They are deciding by two companies: Air Source, Air Conditioning and Prisitine Air Conditioning.

How much money do these companies charge?

In order to make the right decision, they need to know how much each company charges.

Air Source, Air Conditioning:

Charges a fixed value of 50$(service fee)

Charges 10$ per hour

Pristine Air Conditioning:

No service fees

Charges $20 per hour

John's family wants their 4 air conditioners to be installed in every bedroom, so this means that it will take a decent amount of time. They need to know which company is cheaper in different circumstances(If it takes a long time, short time).

Identifying and Classifying the Linear Equations

Assigning the equation of a line (y = mx + b) ( Ax + By = C )

The prices of the air conditioning installation, for each company can be put into the y=mx+b form or the Ax+By=C form.

I will be using the y=mx+b form for representing the cost.

Air Source, Air Conditioning:

y=mx+b (Charge 50$-fixed cost, + 10$ per hour)

y =10x+50

y = Total Cost, M= Cost per hour, X= Amount of Hours, B= Fixed Value(Fixed Cost)

Pristine Air Conditioning:

y=mx + b (Charges no service fee, only 20$ per hour for installation)

y =20x + 0

y = Total Cost, M= Cost per hour, X= Amount of Hours, B= Fixed Value(Fixed Cost)

Table of Values

This table of values, is comparing the cost relative to the hour between the two air Conditioning companies: Air Source, Air Conditioning & Pristine Air Conditioning
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The Graph

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What does the graph consist of?

As you can see, the graph is comparing the total cost relative to the hour between the two companies.

The slopes are represented by the (m-cost per hour) variable in the line equation (y=mx+b).

The intial cost(reference point) are represented by the (b) variable in the line equation.

On the graph that was created, John's Family can see which company is better, for different circumstances.(Within 10 hour maximum)

Finding and Calculating the Point of Intersection

By looking at our graph that was created, we can find the point of intersection

(break-even point). After examining, we can see that the intersection happens at exactly (5,100). The point of intersection represents a point in our graph where both companies charge an even amount of money, for the same amount of hours.

This means that Air Source, Air Conditioning, and Pristine Air Conditioning both charge 100$ for 5 hours.

To prove that our point of intersection is correct we can check algebraically.

Our 2 equations:

y = 10x + 50

y = 20x

Since we want to find the x, we need to make sure both of the equations

(on the right side) equal each other.

10x + 50 = 20x

50 = 20x - 10x

50 = 10x

50/10 = 10x/10


Now we know that 5 is our point of insterction for our x value, we can simply plug this number into our equation to find the y value.

1st Equation:

y =10x + 50

y =10*5 + 50

y =50 + 50

y = 100

2nd Equation:
y = 20x

y = 20*5

y = 100

After checking our answer, we now know that our investigation was correct, and the point of intersection on our graph is exactly (5,100)

Signifigance and Representation of my Linear System

After many calculations and observations we now know which company the Johns family should choose, depending on how many hours it will take. On our graph we see that before our point of intersection (5,100) the company Air Source, Air Conditioning is more expensive then Pristine Air Conditioning.

During the time period of 5 hours, both companies charge 100$. If the job will take longer than 5 hours, then it is better to go with Air Source, Air Conditioning because it will be cheaper compared to Pristine Air Conditioning.

In conclusion, the company Air Source, Air Conditioning is the better way to go, if the job will take less than 5 hours. The company Pristine Air Conditioning is the better way to go, if the job will take more than 5 hours.

Linear systems can help us make the right decisions when we compare different things to find the right choice.

Finding the point of intersection(break even point) is very useful because it tells us when both companies charge an even amount of money for the same amount of service. Before the break even point, it shows which company is more expensive and which company is cheaper. After the break even point it also shows the same thing, but it may change between the companies etc.

John's family may take into consideration the reliability, and reviews of each company so they know their choice will be a good choice even if it is the more expensive decision.

All these factors behind the linear system that was created should go into consideration for Johns family, and assist them in making the best decision.

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Application to real life

This can be a hard decision for John's Family because each company charges different amounts of money and you need to be smart and try to get the cheapest option possible.

Since they have already bought the units, and need it installed, they also need to take in many factors other than cost. They need to make sure that these are safe companies, and that they have good reviews so they know it will be installed properly.

If they wanted to go Air Source Air Conditioning, they should only go with this company if they know that the job will take 5 hours or more. This is because they charge a service fee of 50$ so this has to be taken into account, since the other company does not.

If they wanted to go with Pristine Air Conditionig, they should only go with this company if they know that the job will take 5 hours or less. This is because they charge no service fee, but 20$ per hour which is 10$ more than the other company, and eventually(after 5 hours) they start to pay more and lose money, when they could of got it cheaper if they went with the opposing company.

It does not matter which company they go with(when taking into account, the cost only) if the job only takes 5 hours, because they both charge 100$ for the 5 hour duration.

If this option comes into play, they may choose a prefered company by reviewers, because when choosing what company to service you, if possible you should always try to go with the company with the best service.

To figure out which company is better for another use (E.x. Painting of the house) they would have to examine the costs of each company, and if possible create a linear system to see exactly which service would benefit their needs.

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By: Jaiden P.

Math Project


2ND Period Class