Certainly! Let's understand and derive the required equation step by step.
1. Initial Amount of Juice:
The teacher starts with a total of \(2\) gallons of juice. Since there are \(16\) cups in a gallon, the initial quantity of juice is:
[tex]\[
2 \text{ gallons} \times 16 \text{ cups/gallon} = 32 \text{ cups}
\][/tex]
Thus, initially, there are \(32\) cups of juice.
2. Juice Per Student:
Each student receives \(\frac{1}{2}\) cup of juice.
3. Number of Students:
Let \(x\) represent the number of students served.
4. Juice Given to Students:
If \(x\) students are each given \(\frac{1}{2}\) cup of juice, the total amount of juice given to the students would be:
[tex]\[
x \times \frac{1}{2} = \frac{x}{2} \text{ cups}
\][/tex]
5. Remaining Juice:
To find the remaining juice after serving \(x\) students, we subtract the total juice given to students from the initial amount of juice. This gives:
[tex]\[
y = 32 \text{ cups} - \frac{x}{2} \text{ cups}
\][/tex]
6. Final Equation:
Therefore, the equation representing the remaining amount of juice, \(y\), after serving \(x\) students is:
[tex]\[
y = 32 - \frac{x}{2}
\][/tex]
This equation will help determine how much juice remains after a given number of students have been served.
