<p>The number of integral ordered pair \((x, y)\) that satisfy the system of equations \(|x + y - 4| = 5\) and \(|x - 3| + |y - 1| = 5\) is/are:</p>
Step-by-Step Solution
Key Concept: Solve each absolute value equation separately to get a set of linear equations, then find their intersection points. The first equation represents two parallel lines, while the second represents a diamond (rhombus) shape, and we need to count all integer lattice points satisfying both.
<p><strong>Step 1: Solve |x + y - 4| = 5</strong></p><p>This gives two cases:</p><p>Case 1: x + y - 4 = 5 → x + y = 9</p><p>Case 2: x + y - 4 = -5 → x + y = -1</p><p></p><p><strong>Step 2: Solve |x - 3| + |y - 1| = 5</strong></p><p>This represents a diamond centered at (3, 1) with vertices at (3±5, 1) and (3, 1±5):</p><p>• Vertices: (8, 1), (-2, 1), (3, 6), (3, -4)</p><p></p><p><strong>Step 3: Find intersections with x + y = 9</strong></p><p>Substitute y = 9 - x into |x - 3| + |y - 1| = 5:</p><p>|x - 3| + |9 - x - 1| = 5</p><p>|x - 3| + |8 - x| = 5</p><p></p><p>For x ≤ 3: (3 - x) + (8 - x) = 5 → 11 - 2x = 5 → x = 3, y = 6</p><p>For 3 < x ≤ 8: (x - 3) + (8 - x) = 5 → 5 = 5 ✓ (valid for 3 < x ≤ 8)</p><p>For x > 8: (x - 3) + (x - 8) = 5 → 2x - 11 = 5 → x = 8, y = 1</p><p></p><p>Integer points on x + y = 9 within the diamond: (3,6), (4,5), (5,4), (6,3), (7,2), (8,1) = <strong>6 points</strong></p><p></p><p><strong>Step 4: Find intersections with x + y = -1</strong></p><p>Substitute y = -1 - x into |x - 3| + |y - 1| = 5:</p><p>|x - 3| + |-1 - x - 1| = 5</p><p>|x - 3| + |-2 - x| = 5</p><p></p><p>For x ≤ -2: (3 - x) + (2 + x) = 5 → 5 = 5 ✓ (valid for x ≤ -2)</p><p>For -2 < x ≤ 3: (3 - x) + (-2 - x) = 5 → 1 - 2x = 5 → x = -2, y = 1</p><p>For x > 3: (x - 3) + (-2 - x) = 5 → -5 = 5 ✗</p><p></p><p>Integer points on x + y = -1 within the diamond: (-2,1) = <strong>1 point only</strong></p><p></p><p><strong>Step 5: Verify and count all solutions</strong></p><p>From x + y = 9: (3,6), (4,5), (5,4), (6,3), (7,2), (8,1) — 6 points</p><p>From x + y = -1: Only (-2,1) satisfies the diamond constraint — 0 additional new points that weren't already counted</p><p></p><p>Total integral ordered pairs: <strong>6</strong></p><p></p><p>∴ Answer: C</p>
Correct Answer: C