Question

Jace is buying cookies and brownies for a party. He needs at least 8 cookies and at least 4 brownies. A cookie costs $3 and a brownie costs $5. The most he can spend is $60. A.
Create a system of linear inequalities in two variables from the description. Define your variables.

Consider each of the following potential solutions. For each potential solution, determine whether it is a viable solution, a nonviable solution, or not a solution.(3 points)
viable solution nonviable solution not a solution
10 cookies and 5 brownies
8.5 cookies and 4 brownies
20 cookies and 0 brownies

Answer 1

Let's define the variables and create the system of linear inequalities based on the problem description:

**Variables:**
- Let \( x \) be the number of cookies Jace buys.
- Let \( y \) be the number of brownies Jace buys.

**Inequalities:**
1. Jace needs at least 8 cookies:
\[
x \geq 8
\]
2. Jace needs at least 4 brownies:
\[
y \geq 4
\]
3. Each cookie costs $3 and each brownie costs $5. Jace can spend a maximum of $60:
\[
3x + 5y \leq 60
\]

Thus, the system of inequalities is:
1. \( x \geq 8 \)
2. \( y \geq 4 \)
3. \( 3x + 5y \leq 60 \)

Now, let's evaluate each potential solution:

1. **10 cookies and 5 brownies**:
\[
x = 10, \, y = 5
\]
- Check the inequalities:
\[
10 \geq 8 \quad (\text{true})
\]
\[
5 \geq 4 \quad (\text{true})
\]
\[
3(10) + 5(5) = 30 + 25 = 55 \leq 60 \quad (\text{true})
\]
- **Conclusion**: This is a viable solution.

2. **8.5 cookies and 4 brownies**:
\[
x = 8.5, \, y = 4
\]
- Check the inequalities:
\[
8.5 \geq 8 \quad (\text{true})
\]
\[
4 \geq 4 \quad (\text{true})
\]
\[
3(8.5) + 5(4) = 25.5 + 20 = 45.5 \leq 60 \quad (\text{true})
\]
- **Conclusion**: This is a viable solution.

3. **20 cookies and 0 brownies**:
\[
x = 20, \, y = 0
\]
- Check the inequalities:
\[
20 \geq 8 \quad (\text{true})
\]
\[
0 \geq 4 \quad (\text{false})
\]
\[
3(20) + 5(0) = 60 + 0 = 60 \quad (\text{true})
\]
- **Conclusion**: This is a nonviable solution due to failing the brownie requirement.

**Summary of Solutions:**
- 10 cookies and 5 brownies: **viable solution**
- 8.5 cookies and 4 brownies: **viable solution**
- 20 cookies and 0 brownies: **nonviable solution**