a) Given two lines y = 2x + 1 and 3x - y = 5, find the intersection point of the two lines.

Welcome back — missed you this week. 10.a) Express 126 and 84 as products of their prime factors. Step 1: Find the prime factors of 126. 126 = 2 × 63 = 2 × 3 × 21 = 2 × 3 × 3 × 7 = 2 × 3^2 × 7 Step 2: Find the prime factors of 84. 84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7 = 2^2 × 3 × 7 The prime factors are: 126 = 2 × 3^2 × 7 84 = 2^2 × 3 × 7 10.b) Hence, find the Highest Common Factor (HCF) of 126 and 84. Step 1: Identify common prime factors and their lowest powers. Common prime factors are 2, 3, and 7. For 2: The lowest power is 2^1 (from 2 × 3^2 × 7). For 3: The lowest power is 3^1 (from 2^2 × 3 × 7). For 7: The lowest power is 7^1 (from both). Step 2: Multiply these common prime factors with their lowest powers. HCF = 2^1 × 3^1 × 7^1 = 2 × 3 × 7 = 42 The HCF of 126 and 84 is 42. --- 1. (i) a) Given two lines y = 2x + 1 and 3x - y = 5, find the intersection point of the two lines. Step 1: Substitute the expression for y from the first equation into the second equation. Given: 1) y = 2x + 1 2) 3x - y = 5 Substitute (1) into (2): 3x - (2x + 1) = 5 Step 2: Solve for x. 3x - 2x - 1 = 5 x - 1 = 5 x = 5 + 1 x = 6 Step 3: Substitute the value of x back into the first equation to find y. y = 2(6) + 1 y = 12 + 1 y = 13 The intersection point is (6, 13). 1. (i) b) Find the distance of the point A (-3, 4) from B (1, 7). Step 1: Use the distance formula d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). Let (x_1, y_1) = (-3, 4) and (x_2, y_2) = (1, 7). d = sqrt((1 - (-3))^2 + (7 - 4)^2) Step 2: Calculate the differences and square them. d = sqrt((1 + 3)^2 + (3)^2) d = sqrt((4)^2 + (3)^2) d = sqrt(16 + 9) d = sqrt(25) Step 3: Find the square root. d = 5 The distance between A and B is 5 units. 1. (i) c) Find the midpoint between L (5, 6) and N (1, 2). Step 1: Use the midpoint formula M = ((x_1 + x_2)/(2), (y_1 + y_2)/(2)). Let (x_1, y_1) = (5, 6) and (x_2, y_2) = (1, 2). M = ((5 + 1)/(2), (6 + 2)/(2)) Step 2: Calculate the sums and divide by 2. M = ((6)/(2), (8)/(2)) M = (3, 4) The midpoint between L and N is (3, 4). 1. (ii) a) Complete the table for y = x^2 - 4x + 3. Step 1: Calculate y for each given x value using the function y = x^2 - 4x + 3. For x = -1: y = (-1)^2 - 4(-1) + 3 = 1 + 4 + 3 = 8 For x = 0: y = (0)^2 - 4(0) + 3 = 0 - 0 + 3 = 3 For x = 1: y = (1)^2 - 4(1) + 3 = 1 - 4 + 3 = 0 For x = 2: y = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1 For x = 3: y = (3)^2 - 4(3) + 3 = 9 - 12 + 3 = 0 For x = 4: y = (4)^2 - 4(4) + 3 = 16 - 16 + 3 = 3 The completed table is: |c|c|c|c|c|c|c| x & -1 & 0 & 1 & 2 & 3 & 4 \\ y & 8 & 3 & 0 & -1 & 0 & 3 \\ 1. (ii) b) Draw the graph of the quadratic and the linear function on the same axes (1cm on both axes). To draw the graph: For the quadratic function y = x^2 - 4x + 3: Plot the points from the completed table: (-1, 8), (0, 3), (1, 0), (2, -1), (3, 0), (4, 3). Draw a smooth curve through these points. For the linear function y = x - 1: Choose a few x values and calculate y. For example: If x = 0, y = 0 - 1 = -1. Plot (0, -1). If x = 1, y = 1 - 1 = 0. Plot (1, 0). If x = 4, y = 4 - 1 = 3. Plot (4, 3). Draw a straight line through these points. 1. (ii) c) Mark roots and the y-intercepts of the curve and the line. Roots of the curve (y = x^2 - 4x + 3): These are the x-values where y=0. From the table, the roots are x=1 and x=3. Mark the points (1, 0) and (3, 0) on your graph. Y-intercept of the curve (y = x^2 - 4x + 3): This is the y-value when x=0. From the table, the y-intercept is y=3. Mark the point (0, 3) on your graph. Y-intercept of the line (y = x - 1): This is the y-value when x=0. For y=x-1, when x=0, y=-1. Mark the point (0, -1) on your graph. 1. (ii) d) Using the graph, state the values of x for which x^2 - 4x + 3 = x - 1. Step 1: This equation represents the intersection points of the quadratic curve and the linear line. From the graph, observe where the curve and the line intersect. Alternatively, solve the equation algebraically: x^2 - 4x + 3 = x - 1 Step 2: Rearrange the equation to form a standard quadratic equation. x^2 - 4x - x + 3 + 1 = 0 x^2 - 5x + 4 = 0 Step 3: Factorize the quadratic equation. (x - 1)(x - 4) = 0 Step 4: Solve for x. x - 1 = 0 x = 1 x - 4 = 0 x = 4 The values of x for which x^2 - 4x + 3 = x - 1 are x=1 and x=4. Send me the next one 📸