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!