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 for Question 19. Question 19: Two lines L_1: 2y - 3x - 6 = 0 and L_2: 3y + x = 20 intersect at a point M. Part a) Find the coordinates of M. Step 1: Write the equations in a standard form. L_1: 2y - 3x = 6 (Equation 1) L_2: 3y + x = 20 (Equation 2) Step 2: Solve the system of equations. From Equation 2, express x in terms of y: x = 20 - 3y Substitute this expression for x into Equation 1: 2y - 3(20 - 3y) = 6 2y - 60 + 9y = 6 11y - 60 = 6 11y = 66 y = (66)/(11) y = 6 Substitute the value of y back into the expression for x: x = 20 - 3(6) x = 20 - 18 x = 2 The coordinates of M are (2, 6). Part b) A third line L_3 is perpendicular to L_2 at M. Find the equation of L_3 in the form y = mx + c where m and c are constants. Step 1: Find the gradient of L_2. Rewrite L_2: 3y + x = 20 in the form y = mx + c: 3y = -x + 20 y = -(1)/(3)x + (20)/(3) The gradient of L_2 (m_2) is -(1)/(3). Step 2: Find the gradient of L_3. Since L_3 is perpendicular to L_2, the product of their gradients is -1. m_3 × m_2 = -1 m_3 × (-(1)/(3)) = -1 m_3 = 3 Step 3: Find the equation of L_3. L_3 passes through point M (2, 6) and has a gradient of 3. Using the point-slope form y - y_1 = m(x - x_1): y - 6 = 3(x - 2) y - 6 = 3x - 6 y = 3x - 6 + 6 y = 3x The equation of L_3 is y = 3x. Part c) Another line L_4 is parallel to L_1 and passes through (-1, 3). Determine the x and y intercepts of L_4. Step 1: Find the gradient of L_1. Rewrite L_1: 2y - 3x - 6 = 0 in the form y = mx + c: 2y = 3x + 6 y = (3)/(2)x + 3 The gradient of L_1 (m_1) is (3)/(2). Step 2: Find the gradient of L_4. Since L_4 is parallel to L_1, their gradients are equal. The gradient of L_4 (m_4) is (3)/(2). Step 3: Find the equation of L_4. L_4 passes through (-1, 3) and has a gradient of (3)/(2). Using the point-slope form y - y_1 = m(x - x_1): y - 3 = (3)/(2)(x - (-1)) y - 3 = (3)/(2)(x + 1) y - 3 = (3)/(2)x + (3)/(2) y = (3)/(2)x + (3)/(2) + 3 y = (3)/(2)x + (3)/(2) + (6)/(2) y = (3)/(2)x + (9)/(2) Step 4: Determine the x and y intercepts of L_4. For the y-intercept, set x = 0: y = (3)/(2)(0) + (9)/(2) y = (9)/(2) The y-intercept is (0, (9)/(2)) or (0, 4.5). For the x-intercept, set y = 0: 0 = (3)/(2)x + (9)/(2) -(9)/(2) = (3)/(2)x Multiply both sides by 2: -9 = 3x x = (-9)/(3) x = -3 The x-intercept is (-3, 0). That's 2 down. 3 left today — send the next one.