Find the coordinates of the intercepts and the gradients of the lines 4x - 2y = 8 and x + y = -1, and their intersection point E.

Morning overflow — let's get this done. 1. Solving for lines 4x - 2y = 8 and x + y = -1 a) The coordinates of A and B Line: 4x - 2y = 8 Step 1: Find the x-intercept (point A) by setting y=0. 4x - 2(0) = 8 4x = 8 x = (8)/(4) x = 2 So, point A is (2, 0). Step 2: Find the y-intercept (point B) by setting x=0. 4(0) - 2y = 8 -2y = 8 y = (8)/(-2) y = -4 So, point B is (0, -4). The coordinates are A(2, 0) and B(0, -4). b) The coordinates of C and D Line: x + y = -1 Step 1: Find the x-intercept (point D) by setting y=0. x + 0 = -1 x = -1 So, point D is (-1, 0). Step 2: Find the y-intercept (point C) by setting x=0. 0 + y = -1 y = -1 So, point C is (0, -1). The coordinates are C(0, -1) and D(-1, 0). c) The gradient of 4x - 2y = 8 Step 1: Rearrange the equation into the form y = mx + c. 4x - 2y = 8 -2y = -4x + 8 y = (-4x + 8)/(-2) y = 2x - 4 The gradient m is the coefficient of x. The gradient is 2. d) The gradient of x + y = -1 Step 1: Rearrange the equation into the form y = mx + c. x + y = -1 y = -x - 1 The gradient m is the coefficient of x. The gradient is -1. e) The coordinates of E Point E is the intersection of 4x - 2y = 8 and x + y = -1. Step 1: Use substitution or elimination. From the second equation, y = -x - 1. Substitute this into the first equation: 4x - 2(-x - 1) = 8 4x + 2x + 2 = 8 6x + 2 = 8 6x = 8 - 2 6x = 6 x = 1 Step 2: Substitute x=1 back into y = -x - 1. y = -(1) - 1 y = -2 The coordinates of E are (1, -2). f) The values of x for which the lines are increasing or decreasing Step 1: Determine if the gradient of each line is positive or negative. For 4x - 2y = 8, the gradient is 2 (from part c). Since 2 > 0, this line is increasing. For x + y = -1, the gradient is -1 (from part d). Since -1 < 0, this line is decreasing. The line 4x - 2y = 8 is increasing for all values of x. The line x + y = -1 is decreasing for all values of x. Line 4x - 2y = 8 is increasing for all x. Line x + y = -1 is decreasing for all x. 2. Given: 3x - y = 4 and 2x - y = 5 a) On the same set of axes, draw neat sketch graphs of the two functions. To sketch the graphs, we find two points for each line, typically the intercepts. For 3x - y = 4: If x=0, then -y=4 y=-4. Point: (0, -4). If y=0, then 3x=4 x=(4)/(3). Point: ((4)/(3), 0). For 2x - y = 5: If x=0, then -y=5 y=-5. Point: (0, -5). If y=0, then 2x=5 x=(5)/(2). Point: ((5)/(2), 0). To draw the sketch, plot these points for each line and connect them. The lines will intersect at the point found in part (b). b) Determine the coordinates of the point of intersection. We have the system of equations: 1) 3x - y = 4 2) 2x - y = 5 Step 1: Subtract equation (2) from equation (1) to eliminate y. (3x - y) - (2x - y) = 4 - 5 3x - y - 2x + y = -1 x = -1 Step 2: Substitute x=-1 into either equation. Using equation (1): 3(-1) - y = 4 -3 - y = 4 -y = 4 + 3 -y = 7 y = -7 The coordinates of the point of intersection are (-1, -7). 3. Determine the equations of the following lines in the form f(x) = ax + q. The image shows a line passing through (-3, 0) and (0, 6). The form f(x) = ax + q is equivalent to y = mx + c, where a is the gradient and q is the y-intercept. Step 1: Find the gradient a using the two points (-3, 0) and (0, 6). a = (y_2 - y_1)/(x_2 - x_1) = (6 - 0)/(0 - (-3)) a = (6)/(3) a = 2 Step 2: Identify the y-intercept q. The line crosses the y-axis at (0, 6), so q=6. Step 3: Write the equation in the form f(x) = ax + q. f(x) = 2x + 6 The equation of the line is f(x) = 2x + 6. What's next? 📸