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's the solution to the problem. Question 3: The gradient of the line joining points M(4, 3x) and N(-2x, 1) is (2)/(3). a) Find the value of x. Step 1: Use the gradient formula. The gradient m of a line joining two points (x_1, y_1) and (x_2, y_2) is given by m = (y_2 - y_1)/(x_2 - x_1). Given points M(4, 3x) and N(-2x, 1), and the gradient m = (2)/(3). Let (x_1, y_1) = (4, 3x) and (x_2, y_2) = (-2x, 1). Step 2: Substitute the coordinates and gradient into the formula and solve for x. (2)/(3) = (1 - 3x)/(-2x - 4) Cross-multiply: 2(-2x - 4) = 3(1 - 3x) -4x - 8 = 3 - 9x Add 9x to both sides: -4x + 9x - 8 = 3 5x - 8 = 3 Add 8 to both sides: 5x = 3 + 8 5x = 11 x = (11)/(5) x = 2.2 The value of x is 2.2. b) Correct to three significant figures, |MN|. Step 3: Determine the coordinates of M and N using the value of x. Substitute x = 2.2 into the coordinates of M and N: M(4, 3x) = M(4, 3 × 2.2) = M(4, 6.6) N(-2x, 1) = N(-2 × 2.2, 1) = N(-4.4, 1) Step 4: Use the distance formula to find |MN|. The distance |MN| between two points (x_1, y_1) and (x_2, y_2) is given by sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). |MN| = sqrt((-4.4 - 4)^2 + (1 - 6.6)^2) |MN| = sqrt((-8.4)^2 + (-5.6)^2) |MN| = sqrt(70.56 + 31.36) |MN| = sqrt(101.92) |MN| ≈ 10.0955436 Step 5: Round the distance to three significant figures. |MN| ≈ 10.1 The length of |MN| correct to three significant figures is 10.1. Drop the next question! 📸