To solve the given problem where $y = x^2 - 2x$ and $y = x$, we need to find the points of intersection between the two equations. This involves solving the system of equations by substitution.

Step 1: Substitute $y = x$ into $y = x^2 - 2x$

$x = x^2 - 2x$

Step 2: Rearrange into standard quadratic form

$x^2 - 3x = 0$

Step 3: Factorize the quadratic equation

$x(x - 3) = 0$

Step 4: Solve for $x$

$x = 0 \quad \text{or} \quad x = 3$

Step 5: Find corresponding $y$-values

Using $y = x$, the corresponding $y$-values are:

• For $x = 0$: $y = 0$
• For $x = 3$: $y = 3$

Final Answer:

The points of intersection are: $(0, 0) \quad \text{and} \quad (3, 3)$

Would you like a detailed graph of the solution or further explanation? Here are some related questions to explore:

1. How can we graphically confirm the points of intersection between the two curves?
2. What is the geometric relationship between the parabola $y = x^2 - 2x$ and the line $y = x$?
3. How do we solve similar problems involving quadratic and linear systems of equations?
4. How does factoring simplify the solution process in such problems?
5. What other methods (e.g., completing the square) can we use to solve this?

Tip: Always double-check your factored terms to ensure they simplify the equation correctly.