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.
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Here are the solutions for the requested problems. 1.a) Determine the equation of the line with gradient -3 and passing through (-1, 2). Step 1: Use the point-slope form of a linear equation, y - y_1 = m(x - x_1). Given m = -3, x_1 = -1, y_1 = 2. y - 2 = -3(x - (-1)) y - 2 = -3(x + 1) Step 2: Distribute the gradient and solve for y. y - 2 = -3x - 3 y = -3x - 3 + 2 y = -3x - 1 The equation of the line is y = -3x - 1. 1.c) Determine the equation of the line with gradient -1 and passing through ((1)/(2), -5). Step 1: Use the point-slope form of a linear equation, y - y_1 = m(x - x_1). Given m = -1, x_1 = (1)/(2), y_1 = -5. y - (-5) = -1(x - (1)/(2)) y + 5 = -x + (1)/(2) Step 2: Solve for y. y = -x + (1)/(2) - 5 y = -x + (1)/(2) - (10)/(2) y = -x - (9)/(2) The equation of the line is y = -x - (9)/(2). 2.a) Determine the equation of the line through the points (-1, 7) and (-2, -5). Step 1: Calculate the gradient m using the formula m = (y_2 - y_1)/(x_2 - x_1). Let (x_1, y_1) = (-1, 7) and (x_2, y_2) = (-2, -5). m = (-5 - 7)/(-2 - (-1)) m = (-12)/(-2 + 1) m = (-12)/(-1) m = 12 Step 2: Use the point-slope form y - y_1 = m(x - x_1) with m = 12 and point (-1, 7). y - 7 = 12(x - (-1)) y - 7 = 12(x + 1) Step 3: Distribute the gradient and solve for y. y - 7 = 12x + 12 y = 12x + 12 + 7 y = 12x + 19 The equation of the line is y = 12x + 19. 2.c) Determine the equation of the line through the points (5, 0) and (-3, 2). Step 1: Calculate the gradient m using the formula m = (y_2 - y_1)/(x_2 - x_1). Let (x_1, y_1) = (5, 0) and (x_2, y_2) = (-3, 2). m = (2 - 0)/(-3 - 5) m = (2)/(-8) m = -(1)/(4) Step 2: Use the point-slope form y - y_1 = m(x - x_1) with m = -(1)/(4) and point (5, 0). y - 0 = -(1)/(4)(x - 5) y = -(1)/(4)x + (5)/(4) The equation of the line is y = -(1)/(4)x + (5)/(4). 3 done, 2 left today. You're making progress.