Solve the following system by the addition method.

To solve the system of equations using the addition method (also known as the elimination method), we first write the system again:

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

We already expanded and simplified the second equation in the previous step:

1. \(x^2 + y^2 = 40\)
2. \(x^2 - 10x + 25 + y^2 = 45\)

Now, we subtract equation 1 from equation 2 to eliminate \(y^2\):

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

Simplifying this, we get:

\[
-10x + 25 = 5
\]

Solving for \(x\):

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

Now that we have \(x = 2\), we substitute this value back into equation 1 to find \(y\):

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

Thus, 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 further details on any specific part or have any additional questions?

Here are some related questions for further practice:
1. Solve the system of equations \(x^2 + y^2 = 25\) and \((x - 3)^2 + y^2 = 20\) using the addition method.
2. Find the intersection points of the circles \(x^2 + y^2 = 16\) and \((x + 4)^2 + y^2 = 25\) by elimination.
3. Use the addition method to solve the system \(x^2 + y^2 = 49\) and \((x - 7)^2 + y^2 = 100\).
4. Solve the system \(x^2 + y^2 = 9\) and \(x^2 + (y - 4)^2 = 25\) using the elimination method.
5. Determine the intersection points for the equations \(x^2 + y^2 = 13\) and \((x - 2)^2 + (y - 3)^2 = 10\) by elimination.
6. Find the points where the circles \(x^2 + y^2 = 10\) and \((x + 5)^2 + y^2 = 26\) intersect using the addition method.
7. Solve the system of nonlinear equations \(x^2 + y^2 = 30\) and \((x + 3)^2 + y^2 = 49\) using elimination.
8. Use the addition method to solve \(x^2 + y^2 = 20\) and \((x - 4)^2 + y^2 = 36\).

**Tip:** When using the addition (elimination) method, carefully choose which variable to eliminate first to simplify the equations.

Learn how to solve a system of equations using the addition method (elimination method). This problem involves two equations, x^2 + y^2 = 40 and (x - 5)^2 + y^2 = 45, and requires step-by-step elimination to find the solutions. Suitable for high school students studying algebra and systems of equations.

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