Question

Darnell is buying salad and pizza for a company lunch. Suppose that a bowl of salad costs $5.00, and a slice of pizza costs $3.00.

Let E be the amount in dollars that Darnell spends on salad and pizza. If Darnell buys S bowls of salad and P slices of pizza, then the total
amount of money he spends (E) can be represented by the equation

Now rearrange the equation you wrote above so that P is written in terms of E and S. The quantity of pizza he buys can be represented by the
equation

Suppose Darnell has $30.00 to spend on salad and pizza; that is, E = $30.00.

Complete the following table with the values of S or P that make the equation true.

Hint: To complete the first row, determine the number of pizza slices Darnell can purchase with $30.00, when the number of salad bowls he
purchases is 0.

Budget
(Dollars)
30.00

30.00

30.00

Salad
(Bowls)

0

Pizza
(Slices)

3

0

Answer 1

To formulate the spending equation for Darnell, we start with the cost of salad and pizza:

Let:
- \( S \) = number of bowls of salad
- \( P \) = number of slices of pizza
- \( E \) = total amount spent (in dollars)

The equation for total spending \( E \) can be represented as:

\[
E = 5S + 3P
\]

We want to rearrange this equation to express \( P \) in terms of \( E \) and \( S \):

\[
3P = E - 5S
\]

Dividing both sides by 3 gives us:

\[
P = \frac{E - 5S}{3}
\]

Next, using \( E = 30.00 \) (Darnell's budget), we can rewrite the equation to find \( P \):

\[
P = \frac{30 - 5S}{3}
\]

Now, we can fill out the table using this equation by substituting different values for \( S \) (number of salad bowls) and calculating the corresponding \( P \) (number of pizza slices) values.

1. **When \( S = 0 \)**:
\[
P = \frac{30 - 5(0)}{3} = \frac{30}{3} = 10
\]

2. **When \( S = 1 \)**:
\[
P = \frac{30 - 5(1)}{3} = \frac{30 - 5}{3} = \frac{25}{3} \approx 8.33 \quad \text{(not a whole number, so can't purchase this}
\]

3. **When \( S = 2 \)**:
\[
P = \frac{30 - 5(2)}{3} = \frac{30 - 10}{3} = \frac{20}{3} \approx 6.67 \quad \text{(not a whole number)}
\]

4. **When \( S = 3 \)**:
\[
P = \frac{30 - 5(3)}{3} = \frac{30 - 15}{3} = \frac{15}{3} = 5
\]

5. **When \( S = 4 \)**:
\[
P = \frac{30 - 5(4)}{3} = \frac{30 - 20}{3} = \frac{10}{3} \approx 3.33 \quad \text{(not a whole number)}
\]

6. **When \( S = 5 \)**:
\[
P = \frac{30 - 5(5)}{3} = \frac{30 - 25}{3} = \frac{5}{3} \approx 1.67 \quad \text{(not a whole number)}
\]

7. **When \( S = 6 \)**:
\[
P = \frac{30 - 5(6)}{3} = \frac{30 - 30}{3} = \frac{0}{3} = 0
\]

Now, let's summarize the valid combinations into a table:

| Budget (Dollars) | Salad (Bowls) | Pizza (Slices) |
|-------------------|----------------|-----------------|
| 30.00 | 0 | 10 |
| 30.00 | 3 | 5 |
| 30.00 | 6 | 0 |

These are the valid combinations Darnell can choose within his budget of $30.00. Note that not every increment of salad will yield a whole number of pizza slices based on the budget constraint.