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 $(3, - 6)$ and $(3, 6)$

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 $(3, - 6)$ and $(3, 6)$