Select the correct locations on the image.

Sean decides to start a small business creating and selling outdoor yard games. It will cost Sean [tex] \$50 [/tex] to make each game, as well as an initial cost of [tex] \$300 [/tex] to purchase the needed equipment and supplies. He plans to sell each game for [tex] \$85 [/tex].

The system of equations below models the cost and revenue for Sean's outdoor yard games, where [tex] x [/tex] represents the number of games and [tex] y [/tex] represents the amount in dollars:

[tex]\[
\begin{array}{l}
y = 50x + 300 \\
y = 85x
\end{array}
\][/tex]

1. Select the point on the graph that represents Sean's break-even point, which is the point where the cost to make the games equals the revenue from selling them. Notice that the revenue equation has already been graphed.

2. Determine the least number of games Sean will need to sell in order to make a profit. Profit occurs when revenue is greater than the cost.

Alright, let’s walk through Sean’s business scenario step-by-step.

### Break-Even Point Analysis

Sean has two equations given:
1. Cost Equation: [tex]$ y = 50x + 300 $[/tex]
2. Revenue Equation: [tex]$ y = 85x $[/tex]

These equations will intersect at the break-even point, where the cost matches the revenue. To find this intersection:

1. Set the cost equal to the revenue:
[tex]\[ 50x + 300 = 85x \][/tex]

2. Rearrange the terms to solve for [tex]$ x $[/tex]:
[tex]\[ 85x - 50x = 300 \][/tex]
[tex]\[ 35x = 300 \][/tex]
[tex]\[ x = \frac{300}{35} \][/tex]
[tex]\[ x \approx 8.571428571428571 \][/tex]

Thus, Sean's break-even point happens at approximately 8.57 games.

### Visualizing the Break-Even Point

On a graph where the x-axis represents the number of games (x) and the y-axis represents the dollar amount (y):

- Locate the point where [tex]$ x \approx 8.57 $[/tex]. Since it's not practical to sell a fraction of a game, we consider the closest practical integer.
- The exact coordinate for the break-even (where costs equal revenue) is [tex]$(8.57, \text{ revenue or cost at this point})$[/tex]. However, plotting this might require using rounded values for practical purposes.

### Profit Analysis

To make a profit, revenue must exceed the costs. That means:
[tex]\[ 85x > 50x + 300 \][/tex]
[tex]\[ 35x > 300 \][/tex]
[tex]\[ x > \frac{300}{35} \][/tex]
[tex]\[ x > 8.571428571428571 \][/tex]

Since Sean cannot sell a fraction of a game, he needs to sell at least 9 games to start making a profit.

### Summary

- Break-Even Point: Located at approximately 8.57 games.
- Graphically, you would mark near the (8.57, y) point on the revenue line.
- Profit Starts: Sean will need to sell at least 9 games to make a profit.
- Look for this point (x = 9) on the graph to verify that the revenue exceeds costs.

On the provided image (graph):

1. Break-Even Point: Select at around x = 8.57.
2. Profit Point: Select at x = 9.

With these points identified, Sean can visually and practically understand where he will break even and start making a profit.