Question

Use the situation below to answer both parts of the question.
Quaid and Simone are both selling lemonade. Quaid spent $8 to setup his stand and is charging $2 per glass. Simone only spent $2 on her stand and is charging $1 per glass.

Their situations can be represented by the equations below where P is the total profit and g is the number of glasses sold.

Quaid: P = 2g - 8

Simone: P = 1g - 2

Part 1:

Graph the system that represents Quaid and Simone's lemonade stands to determine when their profit is the same.

Answer 1

To determine when Quaid and Simone's profit is the same, we will graph their profit equations:

1. Quaid's Profit Equation:
\( P = 2g - 8 \)
This means that when he sells \( g \) glasses of lemonade, his profit \( P \) is $2 per glass sold minus his initial setup cost of $8.

2. Simone's Profit Equation:
\( P = 1g - 2 \)
This means that when she sells \( g \) glasses, her profit \( P \) is $1 per glass sold minus her initial setup cost of $2.

### Step 1: Find Points for Each Graph

**For Quaid's equation:**
1. When \( g = 0 \) (selling no glasses):
\( P = 2(0) - 8 = -8 \) → Point (0, -8)

2. When \( g = 6 \):
\( P = 2(6) - 8 = 12 - 8 = 4 \) → Point (6, 4)

3. When \( g = 4 \):
\( P = 2(4) - 8 = 8 - 8 = 0 \) → Point (4, 0)

**For Simone's equation:**
1. When \( g = 0 \) (selling no glasses):
\( P = 1(0) - 2 = -2 \) → Point (0, -2)

2. When \( g = 2 \):
\( P = 1(2) - 2 = 2 - 2 = 0 \) → Point (2, 0)

3. When \( g = 4 \):
\( P = 1(4) - 2 = 4 - 2 = 2 \) → Point (4, 2)

### Step 2: Graph the Equations

- Plot the points for Quaid and Simone on a coordinate plane where the x-axis is the number of glasses sold \( g \), and the y-axis is the profit \( P \).

- **Graphing Quaid’s equation:**
Connect the points (0, -8) and (6, 4). The line should slope upward because as more glasses are sold, the profit increases.

- **Graphing Simone’s equation:**
Connect the points (0, -2) and (4, 2). Similarly, the line should also slope upward.

### Step 3: Determine the Intersection Point

To find when Quaid and Simone's profits are equal (the point where the two lines intersect), we can set their equations equal to each other:

\[ 2g - 8 = g - 2 \]

### Step 4: Solve the Equation

1. Move \( g \) to the left side:
\( 2g - g - 8 = -2 \)
=> \( g - 8 = -2 \)

2. Add 8 to both sides:
\( g = 6 \)

3. Substitute \( g = 6 \) back into either profit equation to find \( P \):
Use Quaid’s equation:
\( P = 2(6) - 8 = 12 - 8 = 4 \)

Thus, the intersection point is (6, 4).

### Conclusion

The two lemonade stands will have the same profit of $4 when they both sell 6 glasses of lemonade. To visualize this, you would plot the two lines on a coordinate graph, where they intersect at the point (6, 4).