We are given the system of equations:

1. \( x^2 - 4x + y^2 = 21 \)
2. \( x - y = 1 \)

### Step 1: Solve for \( y \) in terms of \( x \) using the second equation.
From the second equation, \( x - y = 1 \), solve for \( y \):

\[
y = x - 1
\]

### Step 2: Substitute \( y = x - 1 \) into the first equation.
Now substitute \( y = x - 1 \) into the first equation:

\[
x^2 - 4x + (x - 1)^2 = 21
\]

### Step 3: Expand and simplify the equation.
First, expand \( (x - 1)^2 \):

\[
(x - 1)^2 = x^2 - 2x + 1
\]

Substitute this into the equation:

\[
x^2 - 4x + (x^2 - 2x + 1) = 21
\]

Now, combine like terms:

\[
x^2 + x^2 - 4x - 2x + 1 = 21
\]
\[
2x^2 - 6x + 1 = 21
\]

### Step 4: Move all terms to one side and simplify.
Subtract 21 from both sides:

\[
2x^2 - 6x + 1 - 21 = 0
\]
\[
2x^2 - 6x - 20 = 0
\]

### Step 5: Solve the quadratic equation.
First, simplify the equation by dividing everything by 2:

\[
x^2 - 3x - 10 = 0
\]

Now, solve this quadratic equation. We can factor it:

\[
(x - 5)(x + 2) = 0
\]

Thus, the solutions for \( x \) are:

\[
x = 5 \quad \text{or} \quad x = -2
\]

### Step 6: Solve for \( y \).
Substitute the values of \( x \) into \( y = x - 1 \):

- If \( x = 5 \), then \( y = 5 - 1 = 4 \).
- If \( x = -2 \), then \( y = -2 - 1 = -3 \).

### Final Answer:
The solutions are:
1. \( (x, y) = (5, 4) \)
2. \( (x, y) = (-2, -3) \)

Would you like further details on any step or additional explanations?

Here are some related questions:
1. How do you solve quadratic equations using the quadratic formula?
2. What happens if we have a system of linear equations instead of nonlinear ones?
3. Can we use graphing to solve a system of nonlinear equations?
4. How would you handle a system where the equations involve trigonometric functions?
5. How do you verify solutions to systems of equations?

**Tip:** Always check your solutions by substituting them back into the original equations to ensure they satisfy both equations in the system!

Learn how to solve the system of equations x^2 - 4x + y^2 = 21 and x - y = 1 with a step-by-step solution. This example covers key algebraic techniques, such as substitution, simplification, and solving quadratic equations. Suitable for high school students in Grades 9-11.

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