Prove that for every real number x there is a unique real number y such that x^2y= x−y

We are given the equation \( x^2 y = x - y \), and we want to prove that for every real number \( x \), there is a unique real number \( y \) that satisfies this equation. We'll analyze the different cases for \( x \) and solve for \( y \) accordingly.

### Case 1: \( x = 0 \)
When \( x = 0 \), the equation becomes:
\[
0^2 y = 0 - y
\]
which simplifies to:
\[
0 = -y
\]
Thus, \( y = 0 \).

So, for \( x = 0 \), the unique solution is \( y = 0 \).

### Case 2: \( x \neq 0 \)
When \( x \neq 0 \), the equation is:
\[
x^2 y = x - y
\]
We want to solve this equation

Prove that for every real number x there is a unique real number y such that x^2y = x − y.

Explore how to prove the existence of a unique solution for the equation x^2y = x - y, with detailed case-by-case analysis for both x = 0 and x ≠ 0. This problem is suitable for students in Grades 10-12 and covers essential algebraic techniques and real number properties.

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