Solve the following questions: (a) 2p - 3q = 4, 3p + 2q = 19 (b) 223_x = 87, find x. If x = sqrt((y-z)/(y+z)), find z in terms of x and y.

Here are the solutions to the questions: 1. (a) Solve the simultaneous equations 2p - 3q = 4 (1) 3p + 2q = 19 (2) Step 1: Multiply equation (1) by 2 and equation (2) by 3 to eliminate q. 2(2p - 3q) = 2(4) 4p - 6q = 8 (3) 3(3p + 2q) = 3(19) 9p + 6q = 57 (4) Step 2: Add equation (3) and equation (4). (4p - 6q) + (9p + 6q) = 8 + 57 13p = 65 Step 3: Solve for p. p = (65)/(13) p = 5 Step 4: Substitute p=5 into equation (1). 2(5) - 3q = 4 10 - 3q = 4 -3q = 4 - 10 -3q = -6 q = (-6)/(-3) q = 2 The solution is p = 5 and q = 2. 1. (b) If 223_x = 87, find x. Step 1: Convert the number in base x to base 10. 223_x = 2 · x^2 + 2 · x^1 + 3 · x^0 223_x = 2x^2 + 2x + 3 Step 2: Set the base 10 expression equal to 87. 2x^2 + 2x + 3 = 87 Step 3: Rearrange the equation into a standard quadratic form. 2x^2 + 2x - 84 = 0 Step 4: Divide by 2 to simplify. x^2 + x - 42 = 0 Step 5: Factor the quadratic equation. (x + 7)(x - 6) = 0 Step 6: Solve for x. x + 7 = 0 x = -7 x - 6 = 0 x = 6 Step 7: The base x must be a positive integer greater than the largest digit used (which is 3). Therefore, x = 6. 2. (a) If x = sqrt((y-z)/(y+z)), find z in terms of x and y. Step 1: Square both sides of the equation. x^2 = (y-z)/(y+z) Step 2: Multiply both sides by (y+z). x^2(y+z) = y-z Step 3: Expand the left side. x^2y + x^2z = y-z Step 4: Gather terms containing z on one side and terms without z on the other side. x^2z + z = y - x^2y Step 5: Factor out z. z(x^2 + 1) = y(1 - x^2) Step 6: Solve for z. z = y(1 - x^2)x^2 +