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 are the solutions to your questions: (1) The coordinates of A and B. Step 1: To find the x-intercept (point A), set y=0 in the equation 4x - 2y = 8. 4x - 2(0) = 8 4x = 8 x = (8)/(4) x = 2 So, point A is (2, 0). Step 2: To find the y-intercept (point B), set x=0 in the equation 4x - 2y = 8. 4(0) - 2y = 8 -2y = 8 y = (8)/(-2) y = -4 So, point B is (0, -4). The coordinates of A and B are A(2, 0) and B(0, -4). (2) The coordinates of C and D. Step 1: To find the x-intercept (point D), set y=0 in the equation x + y = -1. x + 0 = -1 x = -1 So, point D is (-1, 0). Step 2: To find the y-intercept (point C), set x=0 in the equation x + y = -1. 0 + y = -1 y = -1 So, point C is (0, -1). The coordinates of C and D are C(0, -1) and D(-1, 0). (3) The gradient of 4x - 2y = 8. Step 1: Rewrite the equation in the slope-intercept form y = mx + c. 4x - 2y = 8 -2y = -4x + 8 y = (-4x)/(-2) + (8)/(-2) y = 2x - 4 The gradient m is the coefficient of x. The gradient of 4x - 2y = 8 is 2. (4) The gradient of x + y = -1. Step 1: Rewrite the equation in the slope-intercept form y = mx + c. x + y = -1 y = -x - 1 The gradient m is the coefficient of x. The gradient of x + y = -1 is -1. (5) The coordinates of E. Step 1: Solve the system of equations for the intersection point E. Equation 1: 4x - 2y = 8 Equation 2: x + y = -1 Step 2: From Equation 2, express y in terms of x: y = -x - 1 Step 3: Substitute this expression for y into Equation 1: 4x - 2(-x - 1) = 8 4x + 2x + 2 = 8 6x + 2 = 8 6x = 8 - 2 6x = 6 x = (6)/(6) x = 1 Step 4: Substitute the value of x back into the expression for y: y = -(1) - 1 y = -2 The coordinates of E are (1, -2). (6) The values of x for which the lines are increasing or decreasing. Step 1: A line is increasing if its gradient is positive, and decreasing if its gradient is negative. For the line 4x - 2y = 8, the gradient is 2 (from part 3). Since 2 > 0, this line is increasing for all values of x. For the line x + y = -1, the gradient is -1 (from part 4). Since -1 < 0, this line is decreasing for all values of x. The line 4x - 2y = 8 is increasing for all x R. The line x + y = -1 is decreasing for all x R. 3 done, 2 left today. You're making progress.