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

### Comparison of ABC and Star Taxi Services

## 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:

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

## Star Taxi Services Did your car break down? Are you getting late to work? What are you going to do? Well, call Star Taxi Services to get to your work on time! Call us at 111-111-STAR! | ## ABC Taxi Services Are you running late to work, instead of driving your car due to a flat tire? How are you going to get to work on time?Call the best taxi service in the world, ABC Taxi Services to get on time! Call us at 222-222-1ABC. | ## Henry needs help, he is confused! Can you help Henry to find out which Taxi services provide a cheap and good deal! |

## Star Taxi Services

Well, call Star Taxi Services to get to your work on time! Call us at 111-111-STAR!

## ABC Taxi Services

**Are you running late to work, instead of driving your car due to a flat tire? How are you going to get to work on time?**

Call the best taxi service in the world, ABC Taxi Services to get on time! Call us at 222-222-1ABC.

## 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**".*

## Taxi Service 1:

## 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

## Taxi Service 2:

## 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

## 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

75-24.75= 50.25

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

## Finding the Point of Intersection

## 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*

*===========================================================================*

__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.

========================================================================

**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)*

## Graph: Point of Intersection

## Comparison of both Taxi Services

## Importance of the Linear System

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.