Solve the following system by the addition method.
\[
\begin{array}{l}
x^{2}+y^{2}=45 \\
(x-7)^{2}+y^{2}=52
\end{array}
\]

Answer

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Answer

The solutions to the system of equations are \(\boxed{(3, -6)}\) and \(\boxed{(3, 6)}\)

Steps

Step 1 ：Given the system of equations: \[x^{2}+y^{2}=45\] and \[(x-7)^{2}+y^{2}=52\]

Step 2 ：Subtract the second equation from the first to eliminate \(y^{2}\), resulting in \[x^{2} - (x - 7)^{2} = -7\]

Step 3 ：Solve the resulting equation for \(x\), yielding \(x = 3\)

Step 4 ：Substitute \(x = 3\) back into the first equation to find \(y\), resulting in two solutions: \(y = -6\) and \(y = 6\)

Step 5 ：The solutions to the system of equations are \(\boxed{(3, -6)}\) and \(\boxed{(3, 6)}\)