To solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] given the equations:
1. [tex]\(\frac{x}{p} + \frac{y}{q} = 2pq\)[/tex]
2. [tex]\(\frac{x}{p^2} + \frac{y}{q^2} = p + q \)[/tex]
Let's work through these step-by-step:
### Step 1: Set up the equations
First, write the given equations in a clearer form:
[tex]\[ \frac{x}{p} + \frac{y}{q} = 2pq \][/tex]
[tex]\[ \frac{x}{p^2} + \frac{y}{q^2} = p + q \][/tex]
### Step 2: Clear denominators by multiplying through
Multiply the first equation by [tex]\(pq\)[/tex]:
[tex]\[ xq + yp = 2p^2q^2 \][/tex]
Multiply the second equation by [tex]\(p^2q^2\)[/tex]:
[tex]\[ xq^2 + yp^2 = p^3q^2 + p^2q^3 \][/tex]
### Step 3: Solve the system of equations using substitution or elimination
Now we have the system:
[tex]\[ \text{(1) } xq + yp = 2p^2q^2 \][/tex]
[tex]\[ \text{(2) } xq^2 + yp^2 = p^3q^2 + p^2q^3 \][/tex]
To solve this system, we first solve for one variable in terms of the other from the first equation. Isolate [tex]\(x\)[/tex] in equation (1):
[tex]\[ xq + yp = 2p^2q^2 \][/tex]
[tex]\[ xq = 2p^2q^2 - yp \][/tex]
[tex]\[ x = \frac{2p^2q^2 - yp}{q} \][/tex]
### Step 4: Substitute the expression for [tex]\(x\)[/tex] into the second equation
Substitute [tex]\(x = \frac{2p^2q^2 - yp}{q}\)[/tex] into equation (2):
[tex]\[ \left(\frac{2p^2q^2 - yp}{q}\right)q^2 + yp^2 = p^3q^2 + p^2q^3 \][/tex]
Simplify:
[tex]\[ \left(2p^2q^2 - yp\right)q + yp^2 = p^3q^2 + p^2q^3 \][/tex]
[tex]\[ 2p^2q^3 - ypq + yp^2 = p^3q^2 + p^2q^3 \][/tex]
Combine like terms and solve for [tex]\(y\)[/tex]:
[tex]\[ yp^2 - ypq = p^3q^2 + p^2q^3 - 2p^2q^3 \][/tex]
[tex]\[ y(p^2 - pq) = p^3q^2 + p^2q^3 - 2p^2q^3 \][/tex]
[tex]\[ y(p^2 - pq) = p^3q^2 - p^2q^3 \][/tex]
We simplify by solving for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{p^3q^2 - p^2q^3}{p^2 - pq} \][/tex]
Similarly, we find that [tex]\(x = p^2q\)[/tex] and [tex]\(y = pq^2\)[/tex].
### Step 5: Calculate [tex]\(x + y\)[/tex]
Add [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x + y = p^2q + pq^2 \][/tex]
Upon simplifying the expression:
[tex]\[ x + y = pq(p + q) \][/tex]
Thus, the value of [tex]\(x + y\)[/tex] is:
[tex]\[ \boxed{pq(p + q)} \][/tex]
This corresponds to option (2).
