# Bus Company Tour Sales

## Situation

A bus company called Magic Tour Bus took a tour bus to Niagara Falls when there were 30 people aboard. The bus company charged the tour bus \$260. The following week, the bus had 50 people on board and the company charged them \$300.

Another bus company called Intercar took a tour bus to Niagara Falls when there were 70 people aboard. The bus charged \$360. The next week the bus, had 120 people aboard and charged them \$510. If you were suggesting someone to buy a tour bus from one of the companies to Niagara Falls, depending on the number of people. Which company would you choose.

## Variables

Let x represent number of people aboard

Let y represent the cost

## Magic Tour Bus Equation

Solution:
Let x = the number of people on the bus
Let y = the total cost for the bus with its passengers.

Then, when there are 30 people on the bus (x = 30), the cost was \$260 (y = 260). This means the point (x, y) = (30, 260).
Also, when there were 50 people on the bus, the cost was \$300. So that is the point (50, 300).

You see - we have two points and so we can write the equation of a line that goes through those points. The first thing we have to do is find the slope.

The slope of the line joining two points is the rise divided by the run. That means

m = (y2 - y1) / (x2 - x1)

So we put in the values of (30, 260) and (50, 300):

m = (300 - 260) / (50 - 30)
m = 40 / 20
m = 2.

The slope is 2. Now we can write it in the form y = mx + b, using 2 for m:

y = 2x + b.

This equation connects both points, so that means it goes through those points. We want to know "b" next, so we can now substitute in either point for the x and the y. I will substitute in x = 50 and y = 220:

300 = 2(50) + b
300 = 100 + b
b + 100 = 300
b = 200.

Now I know the b and the m, so I can finally write the full y=mx + b form of the equation:

y = 2x + 200.

## Variables

Let x represent number of people aboard

Let y represent the cost

## Intercar Tour Bus Equation

Solution:
Let x = the number of people on the bus
Let y = the total cost for the bus with its passengers aboard.

Then, when there are 70 people on the bus (x = 70), the cost was \$360 (y = 360). This means the point (x, y) = (70, 360).
Also, when there were 120 people on the bus, the cost was \$510. So that is the point (120, 510).

You see - we have two points and so we can write the equation of a line that goes through those points. The first thing we have to do is find the slope. That means

m = (y2 - y1) / (x2 - x1)

So we put in the values of (70, 360) and (120, 510):

m = (510 - 360) / (120 - 70)
m = 150 / 50
m = 3.

The slope is 3. Now we can write it in the form y = mx + b, using 3 for m:

y = 3x + b.

This equation connects both points, so that means it goes through those points. We want to know "b" next, so we can now substitute in either point for the x and the y. I will substitute in x = 120 and y = 510:

510 = 3(120) + b
510 = 360 + b
b + 360 = 510
b = 150.

Now I know the b and the m, so I can finally write the full y=mx + b form of the equation:

y = 3x + 150.

## Intersection Point of the Graphs and the solution

According to the graph the point of intersection is (50,300), basically means when 50 people are aboard, the cost of the tour bus is 300 dollars.Another way to show this this is through an equation form, the problem asks how many people aboard will give Magic Tour Bus and Intercar companies the cost, so we want to find x when the y values on the two lines are the same. That is, we want to find where the lines intersect, which we can do by solving the system of equations.

For this system, it's easiest to use substitution. Since

y = 2x=200,

we have,

y=3x+150

2x+200=3x+150

x=50

We think that if 50 people were aboard, the cost will be the same. Let's check this: if Magic Tour Bus has 50 people aboard, the cost is

2(100)+200 = 300,

and if Intercar has 50 people aboard the cost is,

3(50)+150 = 300.

## Summary Statement

If you were suggesting someone to buy a tour bus from one of the companies to Niagara Falls, depending on the number of people. I would suggest using Intercar Tour Bus company if you are taking 50 people aboard or less, but if you are taking more than 50 people aboard, then I suggest you buying Magic Tour Bus company.