# Henry needs a ride to work!!!!!!!!!

## Henry's Dilemma

Henry needs to take a taxi, since his car broke down at Harry Potter Street (15 km away from his house and 35 km away from his office.) He calls his wife, Diana and asks her the price of two taxi companies: Star Taxi Services and ABC Taxi Services. From what Diana said, the following were prices for those two (2) companies:

• Star Taxi Services : Pay \$10.00 per 5 km ( so \$2.00 per 1 km)
• ABC Taxi Services: They charge a fixed cost of \$16.00 and 0.50 cents per 2 km.

## Identifying the Variables

Let the following variables denote the x-value and y-value:

- x value: Distance is represented by the variable "X".

- y-value: Cost is represented by the variable "Y".

## Explanation of Star Taxi Services

• There is no fixed cost (meaning there is no y-intercept), however has a variable cost of \$10.00 per 5 km (\$2.00 per 1 km). This is a direct variation since it has a y-intercept of 0.

## Graph of Star Taxi Services

The x-axis represents the Distance , meanwhile the y-axis denotes the Cost.

## Description of the ABC Taxi Services

• The taxi provider has a fixed cost of \$16.00 and has a variable cost of 0.50 cents per 2 km (0.25 cents per 1 km). This is a partial variation since it has a y-intercept of 16.

## Graph of ABC Taxi Services

The x-axis represents the Distance, meanwhile the y-axis denotes the Cost.

## Which taxi service is cheaper: Star Taxi Services or ABC Taxi Services?

From the given formulas, use them to figure out which one costs cheaper. Remember Henry is 35 km away from his work place .

Use the equations to determine, which Taxi service he needs.

## Solution

Star Taxi Services Answer is:

Use the equation below to determine the cost of driving to Henry's office, which has a distance of 35 km.

y=2x

We know that Henry is currently 35 km away from his work.

Thus, we substitute in the x-value with 35.

y=2 x 35

y= 75

ABC Taxi Services equation is:

Use the equation below to find the cost of driving to Henry's office, which has a distance of 35 km.

y=0.25x +16

We know that Henry is 35 km away from his work place.

Hence we substitute 35 for the x-value.

y=0.25 x 35 +16

y= 8.75 +16

y= 24.75

## Conlusion

Therefore, Henry should call ABC Taxi Services, since ABC Taxi Services has a much cheaper price in comparison to Star Taxi Services. (75>24.75)

75-24.75= 50.25

THERE IS AN EXTENSION TO THIS QUESTION, PLEASE SCROLL DOWN TO SEE THE FULL FLYER!

PART 2

## Solution for Part 2: Finding the Point of Intersection

I will find the P.O.I (Point of Intersection), which will be beneficial when determining which Taxi provider charges the least.

P.O.I

Star Taxi Services Equation in y-intercept form:

y= 2x

ABC Taxi Services Equation in y-intercept form:

y=0.25x+16

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1 Method of finding the Point of Intersection: Solving through graphing

With the prior data, you can use the y-intercept and slope (rise over run).

Here are the steps for the following:

--------------------------------------------------------------------------------------------------------------------------------

Star Taxi Services:

The Equation of the Line:

y=2x

1. You know that there is no y-intercept as mentioned before. Hence starting from the origin (0,0), go 2 grids up and one to the right.

Rise = 2

------------- Therefore, 2/1 gives a slope of 2

Run =1

2. Plot the points: (2,1).

3. Continue with that procedure till you create a line.

ABC Taxi Services:

The Equation of the line:

y=0.25x+16/ y=1/4x+16

1. From the equation, you learned that there is a y-intercept of 16, thus the points will be (0,16).

2.Beginning from the y-intercept of 16 (points: 0,16), use the slope, go 1 grid up and 4 grids to the right.

3. Plot the point: (4,17)

4. Continue with that procedure till you create a line.

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Second Method of finding the P.O.I (Point of Intersection): Solving through substituting

Star Taxi Services:

The Equation of the Line:

y=2x

ABC Taxi Services:

The Equation of the line:

y=0.25x+16/ y=1/4x+16

Equate the equations into one equation

0.25x+16=2x

16=2x-0.25x

16=1.75x

16=1.75x

---- ------------

1.75 1.75

x= 9.14285714286

x= 9.14

Now, to find the y-value, substitute the x-value into one of the equations.

y=0.25x + 16

y= 0.25(9.14) =16

y= 18.2857142857

y= 18.3

Therefore,approximately the point of intersection is:

(9.14,18.3)

## Comparison of both Taxi Services

I would go with the Star Taxi Service before it reaches the P.O.I, since it is much cheaper, however after the P.O.I, I would go with ABC Taxi Services. This taxi service is much cheaper in contrast to Star Taxi Services.

## Importance of the Linear System

A linear system is beneficial in real-life contexts. For instance, for the above scenario, to find out which taxi is cheapest to travel 35 km distance, one can use the linear system. If the distance was less than 9 km, then one could use the star taxi service.

Another important use of linear system is to develop an equation between two variables to see how one variable varies with respect to the other. It is also helpful in making extensions and conclusion.