Algebraic Expression Value Determination

We are given the proportion:

$ \frac{x}{pq - r^2} = \frac{y}{qr - p^2} = \frac{z}{rp - q^2} $

We need to find the value of the expression $px + qy + rz$, given that the denominators are non-zero.

Introducing the Constant Ratio

Let $k$ be the common value of the ratios:

$ k = \frac{x}{pq - r^2} = \frac{y}{qr - p^2} = \frac{z}{rp - q^2} $

From this, we can express $x$, $y$, and $z$ in terms of $k$ and the denominators:

• $ x = k(pq - r^2) $
• $ y = k(qr - p^2) $
• $ z = k(rp - q^2) $

Substituting into the Expression

Now, substitute these expressions for $x$, $y$, and $z$ into the target expression $px + qy + rz$:

$ px + qy + rz = p[k(pq - r^2)] + q[k(qr - p^2)] + r[k(rp - q^2)] $

Simplifying the Result

Factor out the common term $k$:

$ px + qy + rz = k [ p(pq - r^2) + q(qr - p^2) + r(rp - q^2) ] $

Distribute the constants $p$, $q$, and $r$ inside the brackets:

$ px + qy + rz = k [ (p^2q - pr^2) + (q^2r - qp^2) + (r^2p - rq^2) ] $

Rearrange the terms within the brackets to group like terms:

$ px + qy + rz = k [ (p^2q - qp^2) + (q^2r - rq^2) + (r^2p - pr^2) ] $

Simplify the grouped terms:

$ px + qy + rz = k [ 0 + 0 + 0 ] $

$ px + qy + rz = k \times 0 $

$ px + qy + rz = 0 $

Therefore, the value of the expression $px + qy + rz$ is 0.