Edgar was among those responsible for purchasing door prizes for his company's party. Edgar purchased 6 items for $10x each,...

Phase 1: Owning the Dataset

Understanding the Purchase Structure

Let's organize Edgar's purchases:

This gives us the equation:
\(\$60\mathrm{x} + \$30\mathrm{y} = \$600\)

Dividing everything by 30 to simplify:
\(2\mathrm{x} + \mathrm{y} = 20\)

Phase 2: Understanding the Question

We need to find values of x and y from the given choices that jointly satisfy our equation. The key insight: we're looking for two numbers from the list [3, 4, 5, 7, 8, 9] where one represents x and one represents y, such that \(2\mathrm{x} + \mathrm{y} = 20\).

Phase 3: Finding the Answer

Systematic Checking Strategy

We'll test each possible x value and check if the corresponding y value exists in our choices.

If \(\mathrm{x} = 3\):
\(2(3) + \mathrm{y} = 20\)
\(6 + \mathrm{y} = 20\)
\(\mathrm{y} = 14\) (not in our choices)

If \(\mathrm{x} = 4\):
\(2(4) + \mathrm{y} = 20\)
\(8 + \mathrm{y} = 20\)
\(\mathrm{y} = 12\) (not in our choices)

If \(\mathrm{x} = 5\):
\(2(5) + \mathrm{y} = 20\)
\(10 + \mathrm{y} = 20\)
\(\mathrm{y} = 10\) (not in our choices)

If \(\mathrm{x} = 7\):
\(2(7) + \mathrm{y} = 20\)
\(14 + \mathrm{y} = 20\)
\(\mathrm{y} = 6\) (not in our choices)

If \(\mathrm{x} = 8\):
\(2(8) + \mathrm{y} = 20\)
\(16 + \mathrm{y} = 20\)
\(\mathrm{y} = 4\) ✓ (4 IS in our choices!)

Stop here - we found our answer.

Verification

Let's confirm:

- 6 items at \(\$10(8) = 6 \times \$80 = \$480\)
- 4 items at $30 = $120
- Total = $480 + $120 = $600 ✓

Phase 4: Solution

X = 8 and Y = 4

These values satisfy our equation \(2\mathrm{x} + \mathrm{y} = 20\), as \(2(8) + 4 = 16 + 4 = 20\), and result in the correct total purchase amount of $600.