<p>The number of non-negative integral solutions of <span>\(2x + y = 11\)</span> is ______.</p>
Step-by-Step Solution
Key Concept: Recognize this is a linear Diophantine equation where we need to count valid non-negative integer pairs (x,y). For each valid value of x, there's exactly one corresponding y-value, so we only need to find the range of x that keeps y ≥ 0.
<p><strong>Step 1:</strong> Start with the equation 2x + y = 11, where x, y ≥ 0 (non-negative integers).</p><p><strong>Step 2:</strong> Express y in terms of x: y = 11 - 2x</p><p><strong>Step 3:</strong> For y to be non-negative, we need: 11 - 2x ≥ 0 ⟹ 2x ≤ 11 ⟹ x ≤ 5.5</p><p><strong>Step 4:</strong> Since x must be a non-negative integer: x ∈ {0, 1, 2, 3, 4, 5}</p><p><strong>Step 5:</strong> Verify the corresponding y-values:</p><ul><li>x = 0: y = 11 ✓</li><li>x = 1: y = 9 ✓</li><li>x = 2: y = 7 ✓</li><li>x = 3: y = 5 ✓</li><li>x = 4: y = 3 ✓</li><li>x = 5: y = 1 ✓</li></ul><p>∴ Answer: <strong>6</strong></p>
Correct Answer: 6