This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Hey, good to see you again. Step 1: Identify the coordinates of the points from the graph. • Point A: (0, 3) • Point B: (4, 3) • Point C: (4, 0) a) Find the gradient of each of the lines through points A and B, and through points B and C. The formula for the gradient m between two points (x_1, y_1) and (x_2, y_2) is m = (y_2 - y_1)/(x_2 - x_1). For line AB: Points A(0, 3) and B(4, 3). m_AB = (3 - 3)/(4 - 0) = (0)/(4) = 0 The gradient of line AB is 0. For line BC: Points B(4, 3) and C(4, 0). m_BC = (0 - 3)/(4 - 4) = (-3)/(0) Division by zero means the gradient is undefined. This is a vertical line. The gradient of line BC is undefined. b) Find the equation of each of the lines through points A and B, and through points B and C. The formula for the equation of a line is y - y_1 = m(x - x_1). For line AB: Gradient m_AB = 0. Using point A(0, 3). y - 3 = 0(x - 0) y - 3 = 0 y = 3 The equation of line AB is y = 3. For line BC: The line BC is a vertical line passing through x=4. The equation of a vertical line is x = k, where k is the x-coordinate. The equation of line BC is x = 4. c) Find the gradient and equation of the line passing through points A and C. Hence, determine x and y-intercepts. For line AC: Points A(0, 3) and C(4, 0). Gradient m_AC: m_AC = (0 - 3)/(4 - 0) = (-3)/(4) The gradient of line AC is -(3)/(4). Equation of line AC: Using point A(0, 3) and m_AC = -(3)/(4). y - 3 = -(3)/(4)(x - 0) y - 3 = -(3)/(4)x y = -(3)/(4)x + 3 The equation of line AC is y = -(3)/(4)x + 3. Determine x and y-intercepts for line AC: To find the y-intercept, set x = 0: y = -(3)/(4)(0) + 3 y = 3 The y-intercept is (0, 3). To find the x-intercept, set y = 0: 0 = -(3)/(4)x + 3 (3)/(4)x = 3 3x = 12 x = 4 The x-intercept is (4, 0). Send me the next one 📸