Solve the system of equations. \[ \begin{array}{l}y=x^{2}+x-47 \\ y=x+97 \\ \text { Write the coordinates in exact form. Simplify all fractior } \\ (\square), ~\end{array} \]

To solve the system of equations: 1. Set the two equations equal to each other since both equal \(y\): \[ x^2 + x - 47 = x + 97 \] 2. Simplify by subtracting \(x\) from both sides: \[ x^2 - 47 = 97 \] 3. Now subtract 97 from both sides: \[ x^2 - 144 = 0 \] 4. This can be factored as: \[ (x - 12)(x + 12) = 0 \] 5. Thus, the solutions for \(x\) are: \[ x = 12 \quad \text{and} \quad x = -12 \] 6. To find the corresponding \(y\) values, substitute \(x\) back into the simpler equation \(y = x + 97\): For \(x = 12\): \[ y = 12 + 97 = 109 \] For \(x = -12\): \[ y = -12 + 97 = 85 \] 7. Therefore, the solutions in coordinate form are: \[ (12, 109) \quad \text{and} \quad (-12, 85) \] The coordinates in exact form are \( \boxed{(12, 109)}\) and \( \boxed{(-12, 85)}\).