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
Here's the solution to question 19: Part (a): Find the coordinates of M. Step 1: Write the given equations in a standard form. Line L_1: 2y - 3x - 6 = 0 2y - 3x = 6 (Equation 1) Line 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 Now substitute y=6 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. Step 1: Find the gradient of L_2. The equation of L_2 is 3y + x = 20. Rearrange to y = mx + c form: 3y = -x + 20 y = -(1)/(3)x + (20)/(3) The gradient of L_2 is m_2 = -(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: Use the point M (2, 6) and the gradient m_3 = 3 to find the equation of L_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. The equation of L_1 is 2y - 3x - 6 = 0. Rearrange to y = mx + c form: 2y = 3x + 6 y = (3)/(2)x + (6)/(2) y = (3)/(2)x + 3 The gradient of L_1 is m_1 = (3)/(2). Step 2: Find the gradient of L_4. Since L_4 is parallel to L_1, their gradients are equal: m_4 = m_1 = (3)/(2) Step 3: Use the point (-1, 3) and the gradient m_4 = (3)/(2) to find the equation of L_4. Using the point-slope form y - y_1 = m(x - x_1): y - 3 = (3)/(2)(x - (-1)) y - 3 = (3)/(2)(x + 1) Multiply the entire equation by 2 to eliminate the fraction: 2(y - 3) = 3(x + 1) 2y - 6 = 3x + 3 2y = 3x + 9 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) y = 4.5 The y-intercept is (0, 4.5). For the x-intercept, set y = 0: 0 = (3)/(2)x + (9)/(2) Multiply the entire equation by 2: 0 = 3x + 9 -3x = 9 x = (9)/(-3) x = -3 The x-intercept is (-3, 0). Last free one today — make it count tomorrow, or type /upgrade for unlimited.