To solve the given system of equations:

\[ 
\begin{cases}
x^2 + y^2 = 40 \\
(x - 5)^2 + y^2 = 45 
\end{cases}
\]

we need to find the values of \(x\) and \(y\) that satisfy both equations simultaneously.

First, expand the second equation:

\[
(x - 5)^2 + y^2 = 45
\]

Expanding \((x - 5)^2\):

\[
(x - 5)^2 = x^2 - 10x + 25
\]

So the second equation becomes:

\[
x^2 - 10x + 25 + y^2 = 45
\]

Now we have two equations:

\[
\begin{cases}
x^2 + y^2 = 40 \\
x^2 - 10x + 25 + y^2 = 45
\end{cases}
\]

Subtract the first equation from the second equation:

\[
(x^2 - 10x + 25 + y^2) - (x^2 + y^2) = 45 - 40
\]

This simplifies to:

\[
-10x + 25 = 5
\]

Solving for \(x\):

\[
-10x + 25 = 5 \implies -10x = -20 \implies x = 2
\]

Now, substitute \(x = 2\) back into the first equation to solve for \(y\):

\[
2^2 + y^2 = 40 \implies 4 + y^2 = 40 \implies y^2 = 36 \implies y = \pm 6
\]

So, the solutions are:

\[
(x, y) = (2, 6) \text{ or } (2, -6)
\]

Therefore, the solutions to the system are \((2, 6)\) and \((2, -6)\).

Would you like details on any specific part or have any questions?

Here are some related questions for further practice:
1. Solve the system of equations \(x^2 + y^2 = 25\) and \((x - 3)^2 + y^2 = 20\).
2. Find the intersection points of the circles \(x^2 + y^2 = 16\) and \((x + 4)^2 + y^2 = 25\).
3. Determine the coordinates of the points where the ellipse \(x^2 + 4y^2 = 20\) intersects the line \(x - y = 1\).
4. Solve the system \(x^2 + y^2 = 49\) and \((x - 7)^2 + y^2 = 100\).
5. Determine the points of intersection for \(x^2 + y^2 = 9\) and \(x^2 + (y - 4)^2 = 25\).
6. Find the points where the parabola \(y = x^2 - 4\) intersects the circle \(x^2 + y^2 = 16\).
7. Solve the system \(x^2 + y^2 = 13\) and \((x - 2)^2 + (y - 3)^2 = 10\).
8. Determine the intersection points of the circles \(x^2 + y^2 = 10\) and \((x + 5)^2 + y^2 = 26\).

**Tip:** When solving systems of nonlinear equations, it often helps to expand and simplify one of the equations before substituting it into the other.

Learn how to solve a system of nonlinear equations x^2 + y^2 = 40 and (x - 5)^2 + y^2 = 45. Step-by-step solution included with detailed algebraic manipulations. Suitable for advanced high school students and college-level algebra courses.

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