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
To find the equation of a straight line, we first calculate the gradient (m) using the formula m = (y_2 - y_1)/(x_2 - x_1). Then, we use the point-slope form y - y_1 = m(x - x_1) to find the equation. a) Points: (3, 0) and (1, 2) Step 1: Calculate the gradient. Let (x_1, y_1) = (3, 0) and (x_2, y_2) = (1, 2). m = (2 - 0)/(1 - 3) = (2)/(-2) = -1 Step 2: Use the point-slope form with one of the points, e.g., (3, 0), and the gradient m = -1. y - 0 = -1(x - 3) Step 3: Simplify to the slope-intercept form y = mx + c. y = -x + 3 The equation of the line is y = -x + 3. b) Points: (1, 7) and (5, 9) Step 1: Calculate the gradient. Let (x_1, y_1) = (1, 7) and (x_2, y_2) = (5, 9). m = (9 - 7)/(5 - 1) = (2)/(4) = (1)/(2) Step 2: Use the point-slope form with one of the points, e.g., (1, 7), and the gradient m = (1)/(2). y - 7 = (1)/(2)(x - 1) Step 3: Simplify to the slope-intercept form y = mx + c. y - 7 = (1)/(2)x - (1)/(2) y = (1)/(2)x - (1)/(2) + 7 y = (1)/(2)x + (13)/(2) The equation of the line is y = (1)/(2)x + (13)/(2). c) Points: (-1, 2) and (3, 4) Step 1: Calculate the gradient. Let (x_1, y_1) = (-1, 2) and (x_2, y_2) = (3, 4). m = (4 - 2)/(3 - (-1)) = (2)/(3 + 1) = (2)/(4) = (1)/(2) Step 2: Use the point-slope form with one of the points, e.g., (-1, 2), and the gradient m = (1)/(2). y - 2 = (1)/(2)(x - (-1)) y - 2 = (1)/(2)(x + 1) Step 3: Simplify to the slope-intercept form y = mx + c. y - 2 = (1)/(2)x + (1)/(2) y = (1)/(2)x + (1)/(2) + 2 y = (1)/(2)x + (5)/(2) The equation of the line is y = (1)/(2)x + (5)/(2). d) Points: (4, 6) and (10, 9) Step 1: Calculate the gradient. Let (x_1, y_1) = (4, 6) and (x_2, y_2) = (10, 9). m = (9 - 6)/(10 - 4) = (3)/(6) = (1)/(2) Step 2: Use the point-slope form with one of the points, e.g., (4, 6), and the gradient m = (1)/(2). y - 6 = (1)/(2)(x - 4) Step 3: Simplify to the slope-intercept form y = mx + c. y - 6 = (1)/(2)x - (4)/(2) y - 6 = (1)/(2)x - 2 y = (1)/(2)x - 2 + 6 y = (1)/(2)x + 4 The equation of the line is y = (1)/(2)x + 4. That's 2 down. 3 left today — send the next one.