Answer:
\[r = \frac{\frac{x}{y} - z^2 - \frac{1}{y}}{r - 2z}\]
Explanation:
To solve for the variable "r" in the equation \(1 = -x + y \cdot (r - z)^2\), follow these steps:
Step 1: Expand the equation on the right side by squaring \((r - z)\):
\[1 = -x + y \cdot (r^2 - 2rz + z^2)\]
Step 2: Move the terms without "r" to the left side of the equation:
\[x = y \cdot (r^2 - 2rz + z^2) - 1\]
Step 3: Divide both sides by \(y\):
\[\frac{x}{y} = r^2 - 2rz + z^2 - \frac{1}{y}\]
Step 4: Move the constants to the right side:
\[r^2 - 2rz = \frac{x}{y} - z^2 - \frac{1}{y}\]
Step 5: Factor out \(r\) on the left side:
\[r(r - 2z) = \frac{x}{y} - z^2 - \frac{1}{y}\]
Step 6: Solve for \(r\):
\[r = \frac{\frac{x}{y} - z^2 - \frac{1}{y}}{r - 2z}\]
So, the solution for \(r\) in terms of the other variables is:
\[r = \frac{\frac{x}{y} - z^2 - \frac{1}{y}}{r - 2z}\]