<p>Find the number of solutions of the system of equation \(x + 2y = 6\) and \(|x - 3| = y\).</p>
Step-by-Step Solution
Key Concept: Substitute the absolute value expression y = |x - 3| into the linear equation and solve piecewise by considering x ≥ 3 and x < 3 separately, then verify solutions geometrically as intersection points.
<p><strong>Step 1:</strong> Substitute y = |x - 3| into x + 2y = 6 to get x + 2|x - 3| = 6.</p><p><strong>Step 2:</strong> <strong>Case 1 (x ≥ 3):</strong> |x - 3| = x - 3, so x + 2(x - 3) = 6 → 3x - 6 = 6 → x = 4. Then y = |4 - 3| = 1. Check: 4 + 2(1) = 6 ✓. Solution: (4, 1).</p><p><strong>Step 3:</strong> <strong>Case 2 (x < 3):</strong> |x - 3| = 3 - x, so x + 2(3 - x) = 6 → x + 6 - 2x = 6 → x = 0. Then y = |0 - 3| = 3. Check: 0 + 2(3) = 6 ✓ and 0 < 3 ✓. Solution: (0, 3).</p><p><strong>Step 4:</strong> Both solutions satisfy both original equations.</p><p>∴ Answer: <strong>2</strong></p>
Correct Answer: 2