# Linear System Flyer

### By: Rutvik Patel

My friend wants to buy a cellular plan from 2 promotional offers this Christmas. The first offer has a fixed (inital) cost of \$5 and every minute of use costs him \$0.50. The other offer has no fixed (initial) cost but costs \$1 for every minute used.

## Variables

Independent Variable - Number of Minutes

Dependent Variable - Total Cost (in \$)

## Which Offer is Better

To find which offer is better, we must convert the given data into slope intercept form/ standard form, or graph and see which plan is better for each situation.

## Cellular Plan #1

To find the slope

=rise/run

=(5.50-5.00)/(1-0)

=0.5/1

=0.5

Therefore the slope is 0.5.

The y intercept is 5, as that is where the line touches the y-axis.

Therefore the equation for slope intercept form is:

y=0.5x+5

where y is the total cost and x is the number of minutes.

To change into Standard Form:

Ax+By+C=0

0.5x-y+5=0

## Cellular Plan #2

To find the slope:

= rise/run

= (1-0)/(1-0)

=1/1

=1

Therefore the slope is 1.

The y intercept is 0, as that is where the line intersects the y- axis.

Therefore the equation in Slope Intercept Form is:

y=x

To convert into Standard Form:

Ax+By+C=0

x-y=0

## Point of Intersection

To find the point of intersection, we must look at the 2 equations.

For plan #1: y=0.5x+5

For plan #2: y=x

So:

0.5x+5=x

5=x-0.5x

0.5x=5

x=10

Now we subsitute the value of x into the 1 of the equations:

y=0.5*10+5

y=5+5

y=10

To double check we subsitute x into the other equation:

y=10

Now that both of y's are the equal we know that the y point is correct.

Therefore, the point of intersection is (10,10).

We can also double check by looking at the graph.

## Summary

Both Cellular Plans will cost the same amount (\$10) for 10 minutes of usage. This is also the point of intersection.

## Decision

As we can see from the graph Cellular Plan #2 is cheaper if he doesn't exceed 10 minutes. If he likes to use more that 10 minutes than he should chose Cellular Plan #1.

## Significance of Solution

The graph helps compare the costs of various plans in real life as well by chosing an appropriate plan it can save significant dollars for the users.