Solve the simultaneous equations: x+y=5 and x2+y2=13. (b) Find xy

Okay, bazuunelson, let's tackle these. 13. (a) Solve the simultaneous equations: x+y=5 (1) x^2+y^2=13 (2) Step 1: Express y in terms of x from equation (1). From (1), we have y = 5-x. Step 2: Substitute this expression for y into equation (2). x^2 + (5-x)^2 = 13 Step 3: Expand and simplify the equation. x^2 + (25 - 10x + x^2) = 13 2x^2 - 10x + 25 = 13 2x^2 - 10x + 12 = 0 Step 4: Divide the entire equation by 2. x^2 - 5x + 6 = 0 Step 5: Factor the quadratic equation to find the values of x. (x-2)(x-3) = 0 So, x=2 or x=3. Step 6: Find the corresponding values of y using y=5-x. If x=2, then y = 5-2 = 3. If x=3, then y = 5-3 = 2. The solutions are (x,y) = (2,3) and (x,y) = (3,2). The solutions are x=2, y=3 or x=3, y=2. 13. (b) Find xy Step 1: Use the solutions found in part (a). If x=2 and y=3, then xy = 2 × 3 = 6. If x=3 and y=2, then xy = 3 × 2 = 6. In both cases, xy is the same. xy = 6 The value of xy is 6. 14. (a) Find the equation of a line passing through (2,3) and (4,7) Step 1: Calculate the gradient (slope) m of the line. The formula for the gradient is m = (y_2 - y_1)/(x_2 - x_1). Using points (x_1, y_1) = (2,3) and (x_2, y_2) = (4,7): m = (7-3)/(4-2) = (4)/(2) = 2 Step 2: Use the point-slope form of the equation of a line, y - y_1 = m(x - x_1). Using the point (2,3) and m=2: y - 3 = 2(x - 2) Step 3: Simplify the equation to the slope-intercept form y=mx+c. y - 3 = 2x - 4 y = 2x - 4 + 3 y = 2x - 1 The equation of the line is y = 2x - 1. 14. (b) Find the midpoint of the line Step 1: Recall the midpoint formula. For two points (x_1, y_1) and (x_2, y_2), the midpoint M is given by M = ((x_1+x_2)/(2), (y_1+y_2)/(2)). Step 2: Substitute the given points (2,3) and (4,7) into the formula. M = ((2+4)/(2), (3+7)/(2)) Step 3: Calculate the coordinates of the midpoint. M = ((6)/(2), (10)/(2)) M = (3,5) The midpoint of the line is (3,5). That's 2 down. 3 left today — send the next one.