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 $q$.

$$2(2p - 3q) = 2(4) \implies 4p - 6q = 8 \quad (3)$$ $$3(3p + 2q) = 3(19) \implies 9p + 6q = 57 \quad (4)$$

Step 2: Add equation (3) and equation (4).

$$(4p - 6q) + (9p + 6q) = 8 + 57$$ $$13p = 65$$

Step 3: Solve for $p$.

$$p = \frac{65}{13}$$ $$p = 5$$

Step 4: Substitute $p=5$ into equation (1).

$$2(5) - 3q = 4$$ $$10 - 3q = 4$$ $$-3q = 4 - 10$$ $$-3q = -6$$ $$q = \frac{-6}{-3}$$ $$q = 2$$

The solution is $p = \boxed{5}$ and $q = \boxed{2}$.

1. (b) If $223_x = 87$, find $x$.

Step 1: Convert the number in base $x$ to base 10.

$$223_x = 2 \cdot x^2 + 2 \cdot x^1 + 3 \cdot x^0$$ $$223_x = 2x^2 + 2x + 3$$

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 $x$.

$$x + 7 = 0 \implies x = -7$$ $$x - 6 = 0 \implies x = 6$$

Step 7: The base $x$ must be a positive integer greater than the largest digit used (which is 3).
Therefore, $x = \boxed{6}$.

2. (a) If $x = \sqrt{\frac{y-z}{y+z}}$, find $z$ in terms of $x$ and $y$.

Step 1: Square both sides of the equation.

$$x^2 = \frac{y-z}{y+z}$$

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 $z$.

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