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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38 Find the equation of a straight line whose gradient is 3 and passes through points C(-2, 5). Step 1: Identify the given gradient and point. Given gradient m = 3. Given point (x_1, y_1) = (-2, 5). Step 2: Use the point-slope form of a linear equation, y - y_1 = m(x - x_1). Substitute the values: y - 5 = 3(x - (-2)) y - 5 = 3(x + 2) Step 3: Simplify the equation into the form y = mx + c. y - 5 = 3x + 6 Add 5 to both sides: y = 3x + 6 + 5 y = 3x + 11 The equation of the straight line is y = 3x + 11. 39 Use a number line to represent the inequality. The inequality to be represented is not provided in the image. Therefore, it is not possible to represent it on a number line. 40 Find the perimeter of the figure below. Step 1: Assume the figure is a rectangle. For a rectangle, opposite sides are equal. From the diagram, the lengths are (3x+7)m and (4x-10)m, and the width is (2x+5)m. Set the expressions for the length equal to each other to find x: 3x + 7 = 4x - 10 Step 2: Solve for x. Subtract 3x from both sides: 7 = x - 10 Add 10 to both sides: 7 + 10 = x x = 17 Step 3: Calculate the actual length and width of the rectangle. Length L = 3x + 7 = 3(17) + 7 = 51 + 7 = 58 m. Width W = 2x + 5 = 2(17) + 5 = 34 + 5 = 39 m. Step 4: Calculate the perimeter. The perimeter P of a rectangle is given by P = 2(L + W). P = 2(58 + 39) P = 2(97) P = 194 m The perimeter of the figure is 194 m. 41 Solve for y in: (1)/(2)y - 5 = 7. Step 1: Add 5 to both sides of the equation. (1)/(2)y - 5 + 5 = 7 + 5 (1)/(2)y = 12 Step 2: Multiply both sides by 2 to solve for y. 2 × (1)/(2)y = 12 × 2 y = 24 The value of y is 24. 42 Find the gradient of the line shown. Step 1: Identify the coordinates of the two points on the line. The points are (-2,-3) and (8,4). Let (x_1, y_1) = (-2,-3) and (x_2, y_2) = (8,4). Step 2: Use the gradient formula. The gradient m is given by: m = (y_2 - y_1)/(x_2 - x_1) Step 3: Substitute the values and calculate. m = (4 - (-3))/(8 - (-2)) m = (4 + 3)/(8 + 2) m = (7)/(10) The gradient of the line is (7)/(10). 43 Find the gradient of the line shown below points C(-2, -3) and (8, 7). Step 1: Identify the coordinates of the two points. The points are C(-2,-3) and (8,7). Let (x_1, y_1) = (-2,-3) and (x_2, y_2) = (8,7). Step 2: Use the gradient formula. m = (y_2 - y_1)/(x_2 - x_1) Step 3: Substitute the values and calculate. m = (7 - (-3))/(8 - (-2)) m = (7 + 3)/(8 + 2) m = (10)/(10) m = 1 The gradient of the line is 1. That's 2 down. 3 left today — send the next one.