Let 'x' be the number of games played by a participant.

Then Plan 1 cost can be given as: **y1 = 5 + 1 * x = 5+x** (admission fee + cost per game*number of games)

Similarly, for plan 2, cost is: **y2 = 2.5 + 1.5 * x = 2.5+1.5x** (admission fee + cost per game* number of games)

For point of intersection: both equation will be equal to each other

That is, 5+x = 2.5+1.5x

or, 5-2.5 = 1.5x-x

or 2.5 = 0.5 x

which means, x = 2.5/0.5 = 5

and corresponding y will be: 5+x = 10.

We can also check with plan 2: 2.5+1.5*5 = 2.5+7.5 = 10 =Plan 1 cost

Thus, coordinates for point of intersection are **(5,10)**.

For plan 1 to be more cost effective than Plan 2: y1 < y2

or 5+x < 2.5+1.5 x

or 2.5<0.5x

or **x > 5**

i.e. if a participant plays more than 5 games, Plan 1 is better than Plan 2.

lets check for x = 6

y1= 5+6 = $11.

y2 = 2.5 +1.5*6 = 2.5 + 9 = 11.5

i.e. y1<y2 . or plan 1 is better.

A)

The question tells us that **x **represents the number of games played by a person. It also says that **y **is the total cost.

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

**Plan 1:**

We know that the admission fee is $5.00, and that you pay $1.00 per game. So, you pay $1 times **x,** the number of games.

We can write this as:

**y **(total cost) **=** **5** (admission fee) **+** **1x **($1 for every game played)

**y = 5 + 1x**

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

**Plan 2:**

Now, we can do this for the second plan, using the same logic.

The new admission fee is $2.50. However, you have to pay $1.50 per game. Following the way we wrote the first equation:

**y **(total cost) **= 2.50 **(admission fee) **+ 1.50x **($1.50 for every game played)

**y = 2.50 + 1.50x**

B)

You can estimate a set of coordinates here, we'll check if they're right in part C).

C)

We need to solve the system of equations from part A). To solve a system of equations means to find the value of x where both equations are equal. To recap:

**Plan 1: **y = 5 + x

**Plan 2:** y = 2.5 + 1.5x

The easiest way to solve this system of equations is to make an equation using both of the right sides, like so:

5 + x = 2.5 + 1.5x

Let's solve it:

First, let's gather the like terms.

5 + x = 2.5 + 1.5x

5 + x **- x - 2.5 **= 2.5 + 1.5x **- x - 2.5**

5 - 2.5 = 1.5x - x

Now, we're able to simplify this equation so we can move forward:

5 - 2.5 = 1.5x - x

2.5 = 0.5x

Our next step is to isolate the value of x. We use equal operations, like before:

2.5 = 0.5x

2.5 **/ 0.5 **= 0.5x **/ 0.5**

5 = x

Therefore, the x coordinate in the solution for the system is 5.

In a system of equations, the solution is always an (x,y) pair. To get the y value, we just use one equation, like so:

y = 5 + x (Plan 1)

y = 5 + 5

y = 10

Therefore, the y coordinate in the solution is 10.

The solution for this system of equations is (5,10).

D)

We know that when 5 games are played (the x value we found earlier,) the plans have the same value. However, if you play 4 games or less, plan 2 is better. We can demonstrate this using the equations:

If x is 1:

Plan 1: y = 5 + 1 = 6

Plan 2: y = 2.5 + 1.5 = 4

x is 2:

Plan 1: y = 5 + 2 = 7

Plan 2: y = 2.5 + 3 = 5.5

And so on. However, when we hit 5 games, we know that they are equal, and at 6:

Plan 1: y = 5 + 6 = 11

Plan 2: y = 2.5 + 9 = 11.5

Plan 1 becomes better.

Therefore, at 6 games or more, Plan 1 is better than Plan 2.