# The Diamond Hotel VS The Onyx Hotel

## Which Hotel Is a Better Deal?

A couple and two children are going to Niagara Falls during the Christmas break for one night. They are choosing between two different hotels to reside in. The pricing for these hotels are based on the number of children you bring with you. For the Diamond Hotel, the price for a couple without children for one night is \$250. Every additional child is \$50 more. For the Onyx Hotel, the price for a couple without children for one night is \$175. Every additional child is \$75 more.

## The Onyx Hotel

Variables

The two variables that were used in this relation were C and K

Let C represents the cost

Let K represents the number of children

Equations

The Diamond Hotel has a price for a couple for one night for \$250, and every additional child is \$50. Therefore, the equation for this data would be...

y=mx+b

c=50k+250

or

Ax+By+C=0

50k-y+250=0

The Onyx Hotel has a price for a couple for one night for \$175, and every additional child is \$75. Therefore, the equation for this data would be...

y=mx+b

c=75k+175

or

Ax+By+C=0

75x-y+175=0

Finding the Intersecting Point

50k+250=75k+175

50k-75k=175-250

-25k=-75

-25k/-25=-75/-25

k=3

Now that we have solved for k, we can substitute the value back into the original equation to get the second coordinate.

c=50k+250

c=50(3)+250

c=150+250

c=400

c=75k+175

c=75(3)+175

c=225+175

c=400

Therefore, the point of intersection is (3,400)

Solving this linear system was quite significant because it gives us the comparison between the prices of the two hotels. Now, the family can choose the hotel that will benefit them financially.

In conclusion, if the family were to bring less than three children to this trip, the Onyx Hotel would be a better deal, as it costs less. However, if the family were to bring 3 children to this trip, either hotel would be good, as both the prices would be \$400. Finally, if the family were to bring more than three children, the Diamond hotel would be a better deal as it would cost less.

## Which hotel offers a better deal for this family?

Well, since we found the point of intersection, we now know when the two hotels will cost the same (3 children). However, this is a family of four, two adults and two children. Since two falls below the point of intersection on the graph, it indicates that the Onyx Hotel would give them a better deal of \$325 per night as opposed to the Diamond Hotel which offers \$350 per night.