We are given the following system of equations:

$1. \quad x^2 - 10x - y = -25$ $2. \quad -2x^2 + 20x - y = 50$

Step 1: Eliminate $y$ by subtracting the two equations.

We'll subtract Equation 1 from Equation 2 to eliminate $y$.

$(-2x^2 + 20x - y) - (x^2 - 10x - y) = 50 - (-25)$

Simplifying:

$-2x^2 + 20x - y - x^2 + 10x + y = 50 + 25$ $-3x^2 + 30x = 75$

Step 2: Solve for $x$.

The equation simplifies to:

$-3x^2 + 30x = 75$

Divide the entire equation by $-3$:

$x^2 - 10x = -25$

This equation looks familiar—it's actually the same as the left-hand side of Equation 1! So now we know:

$x^2 - 10x = -25$

Step 3: Solve for $y$.

Now substitute $x^2 - 10x = -25$ back into either of the original equations to find $y$. Let’s use Equation 1:

$x^2 - 10x - y = -25$

Substitute $x^2 - 10x = -25$:

$-25 - y = -25$

This simplifies to:

$y = 0$

Step 4: Verify the solution.

Now, substitute $y = 0$ into both original equations to check if the solution holds. Let’s first use Equation 1:

$x^2 - 10x - 0 = -25$ $x^2 - 10x = -25$

This is true, as we already used this in our calculations. Now, use Equation 2:

$-2x^2 + 20x - 0 = 50$

Substitute $x^2 - 10x = -25$:

$-2(-25) = 50$ $50 = 50$

Both equations hold true, so the system is consistent.

Final Answer:

The solution to the system is:

$y = 0, \quad x^2 - 10x = -25$

This represents a quadratic equation in terms of $x$, so the values of $x$ are the solutions to:

$x^2 - 10x + 25 = 0$

Factoring:

$(x - 5)^2 = 0$

Thus, $x = 5$.

Solution:

$x = 5, \quad y = 0$

Would you like more details on this method or have any other questions?

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Here are 5 related questions:

1. Can you solve a system of nonlinear equations graphically?
2. How would you solve a system of equations with more than two variables?
3. What are the different methods to solve systems of equations?
4. What is the substitution method, and how does it compare to elimination?
5. How can systems of equations be used in real-life applications?

Tip: Always check your solution by substituting the values back into the original equations to ensure consistency.