Use elimination to solve the nonlinear system of equations.
$\begin{array}{l}
x^2+y^2=25 \\
x^2+7 y=37
\end{array}$
The objective is to find all real pairs \((x, y)\) that satisfy the following system of nonlinear equations:
\[ \begin{cases} x^2 + y^2 = 25 \quad (1) \ x^2 + 7y = 37 \quad (2) \end{cases} \]
By analyzing the system, we notice that both equations contain the term \(x^2\). This allows us to use the elimination method by subtracting one equation from the other to remove \(x^2\) and reduce the problem to solving for \(y\).
To eliminate \(x^2\), we subtract equation (2) from equation (1):
\[(x^2 + y^2) - (x^2 + 7y) = 25 - 37\]
This simplifies to:
\[y^2 - 7y = -12\]
Adding 12 to both sides gives the quadratic equation:
\[y^2 - 7y + 12 = 0\]
We can factor this quadratic expression:
\[(y - 3)(y - 4) = 0\]
Setting each factor to zero, we find the potential values for \(y\):
\[y = 3 \quad \text{or} \quad y = 4\]
Now, we substitute each value of \(y\) back into one of the original equations to find the corresponding values of \(x\). Let's use equation (2), \(x^2 + 7y = 37\), which implies \(x^2 = 37 - 7y\).
Case 1: If \(y = 3\):
\[x^2 = 37 - 7(3) = 37 - 21 = 16\]
Taking the square root of both sides, we get \(x = 4\) or \(x = -4\). This gives two solution pairs: \((4, 3)\) and \((-4, 3)\).
Case 2: If \(y = 4\):
\[x^2 = 37 - 7(4) = 37 - 28 = 9\]
Taking the square root of both sides, we get \(x = 3\) or \(x = -3\). This gives two more solution pairs: \((3, 4)\) and \((-3, 4)\).
Combining the results, we have identified four pairs \((x, y)\) that satisfy both equations in the system. The solutions are:
\((4, 3)\), \((-4, 3)\), \((3, 4)\), and \((-3, 4)\).
Each pair can be verified:
For \((4, 3)\): \(4^2 + 3^2 = 16 + 9 = 25\) and \(4^2 + 7(3) = 16 + 21 = 37\).
For \((3, 4)\): \(3^2 + 4^2 = 9 + 16 = 25\) and \(3^2 + 7(4) = 9 + 28 = 37\).
The negative cases hold symmetrically since \(x\) is squared.
The solution set is \(\boxed{{(4, 3), (-4, 3), (3, 4), (-3, 4)}}\).
In physics, solving a system like this can represent finding intersection points of paths or balancing conditions. For example, if \(x^2 + y^2 = 25\) represents the circular orbit of an object and \(x^2 + 7y = 37\) represents a trajectory constraint, these solutions represent the specific coordinates in a coordinate plane where the object satisfies both physical constraints simultaneously. This helps engineers and scientists predict object positions or verify design parameters.