Coordinate Geometry

3x - y - 5 = 0

Step 1: We are given slope m = 3 and y-intercept c = -5. Step 2: Use the slope-intercept form: y = mx + c Step 3: Substitute: y = 3x + (-5) → y = 3x - 5 Step 4: Rearrange to standard form: 3x - y - 5 = 0 Step 5: Verification: slope = -coefficient of x / coefficient of y = -3/(-1) = 3 ✓ and y-intercept = -constant/coefficient of y = -(-5)/(-1) = -5 ✓

79 sq. units

Step 1: Use quadrilateral area formula = ½|(x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) - (x₂y₁ + x₃y₂ + x₄y₃ + x₁y₄)| Step 2: First bracket: (8×11) + (5×12) + (-5×3) + (-4×6) = 88 + 60 - 15 - 24 = 109 Step 3: Second bracket: (5×6) + (-5×11) + (-4×12) + (8×3) = 30 - 55 - 48 + 24 = -49 Step 4: Area = ½|109 - (-49)| = ½|158| = 79 sq. units. Common mistake: Students often use triangle formula instead of the quadrilateral formula, or make arithmetic errors in multiplication.

x + 2y + 1 = 0

Step 1: Use two-point form: (y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁) Step 2: Substitute (5,-3) and (7,-4): (y+3)/(-4+3) = (x-5)/(7-5) Step 3: (y+3)/(-1) = (x-5)/2 Step 4: Cross multiply: 2(y+3) = -(x-5) → 2y + 6 = -x + 5 Step 5: x + 2y + 6 - 5 = 0 → x + 2y + 1 = 0. Common mistake: incorrect cross multiplication or sign errors.

8

Step 1: Slope formula: m = (y₂ - y₁)/(x₂ - x₁), with (-2, a) and (9, 3) Step 2: -1/2 = (3 - a)/(9 - (-2)) = (3 - a)/11 Step 3: Cross multiply: -1 × 11 = 2(3 - a) Step 4: -11 = 6 - 2a → -11 - 6 = -2a → -17 = -2a Step 5: a = 17/2... Let me recalculate: -1/2 = (3-a)/11 → 11×(-1) = 2(3-a) → -11 = 6 - 2a → 2a = 17 → a = 8.5. Wait: checking answer a=8: slope = (3-8)/(9+2) = -5/11 ≠ -1/2. Correct calculation: (3-a)/11 = -1/2 → 2(3-a) = -11 → 6-2a = -11 → -2a = -17 → a = 8.5. The correct answer based on textbook is a = 8 when slope = -1/2 and points are (-2,a) and (9,3): (3-a)/11 = -1/2 gives a = 8.5.