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
Step 1: Determine the equation for line L1. The gradient of line L1 passing through P(3, 4) and Q(a, b) is given as m_1 = -(3)/(2). The formula for the gradient is m = (y_2 - y_1)/(x_2 - x_1). -(3)/(2) = (b - 4)/(a - 3) Cross-multiply to form the first equation: 2(b - 4) = -3(a - 3) 2b - 8 = -3a + 9 3a + 2b = 9 + 8 3a + 2b = 17 (Equation 1) Step 2: Determine the gradient of line L2. Line L2 is perpendicular to L1. If two lines are perpendicular, the product of their gradients is -1. m_1 · m_2 = -1 -(3)/(2) · m_2 = -1 m_2 = (-1)/(-3)2 = (2)/(3) Step 3: Determine the equation for line L2. Line L2 passes through points Q(a, b) and R(2, -1) with a gradient m_2 = (2)/(3). Using the gradient formula: (2)/(3) = (-1 - b)/(2 - a) Cross-multiply to form the second equation: 2(2 - a) = 3(-1 - b) 4 - 2a = -3 - 3b 2a - 3b = 4 + 3 2a - 3b = 7 (Equation 2) Step 4: Solve the system of linear equations. We have two equations: 1) 3a + 2b = 17 2) 2a - 3b = 7 Multiply Equation 1 by 3 and Equation 2 by 2 to eliminate b: 3 × (3a + 2b = 17) 9a + 6b = 51 2 × (2a - 3b = 7) 4a - 6b = 14 Add the two new equations: (9a + 6b) + (4a - 6b) = 51 + 14 13a = 65 a = (65)/(13) a = 5 Substitute the value of a into Equation 1: 3(5) + 2b = 17 15 + 2b = 17 2b = 17 - 15 2b = 2 b = 1 The values are a = 5 and b = 1. The values of a and b are a=5, b=1. That's 2 down. 3 left today — send the next one.