https://file.aiquickdraw.com/mathbot/file/1728232830607a0fowasl.jpg

Let's analyze and solve the problem presented in the image.

### Given Equations:
1. \( x + y = p + q \) (Equation 1)
2. \( p x - q y = p^2 - q^2 \) (Equation 2)

We are tasked with solving these equations for \(x\) and \(y\).

### Step-by-step solution:

From **Equation 1**: 
\[
x + y = p + q \implies x = p + q - y \tag{3}
\]

Substitute this expression for \(x\) into **Equation 2**:
\[
p(p + q - y) - qy = p^2 - q^2
\]

Now, expand this:
\[
p(p + q - y) = p(p + q) - p y = p^2 + p q - p y
\]
So, the equation becomes:
\[
(p^2 + pq - py) - qy = p^2 - q^2
\]
Simplify it:
\[
p^2 + pq - py - qy = p^2 - q^2
\]
Cancel \(p^2\) on both sides:
\[
pq - py - qy = -q^2
\]
Factor \(y\) out:
\[
pq - y(p + q) = -q^2
\]
Solve for \(y\):
\[
y = \frac{pq + q^2}{p + q}
\]

Now, substitute this expression for \(y\) into **Equation 1** to find \(x\):
\[
x + \frac{pq + q^2}{p + q} = p + q
\]
Simplify:
\[
x = p + q - \frac{pq + q^2}{p + q}
\]
To simplify this, find a common denominator:
\[
x = \frac{(p + q)(p + q) - (pq + q^2)}{p + q}
\]
Expand and simplify the numerator:
\[
(p + q)^2 - (pq + q^2) = p^2 + 2pq + q^2 - pq - q^2 = p^2 + pq
\]
Thus:
\[
x = \frac{p^2 + pq}{p + q}
\]

### Final Solution:
\[
x = \frac{p^2 + pq}{p + q}, \quad y = \frac{pq + q^2}{p + q}
\]

Would you like any further details or have any questions about these steps?

Here are some related questions:
1. How would the solution change if \(p = q\)?
2. What if \(p\) or \(q\) were zero? How would that affect \(x\) and \(y\)?
3. Could you solve the system using substitution from \(y\) instead of \(x\)?
4. How could these equations be solved using matrices?
5. What real-world applications can this type of system of equations represent?

**Tip**: Always simplify equations carefully when solving systems of equations to avoid unnecessary complications.

Solve the given system of equations: x + y = p + q and px - qy = p^2 - q^2

Learn how to solve the system of linear equations x + y = p + q and px - qy = p^2 - q^2 using substitution and algebraic methods. Step-by-step solution for high school students in grades 10-12.

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