Golve the system of equations, \[ y=x^{3}-24 x+22 \] \( y=-92 x+10 \) Write the coordinates in exact form. Simplify all fractions and radi \( (1) \) )

To solve the system of equations, we will set the two equations equal to each other: \[ x^3 - 24x + 22 = -92x + 10. \] Rearranging gives us: \[ x^3 - 24x + 92x + 22 - 10 = 0, \] which simplifies to: \[ x^3 + 68x + 12 = 0. \] Now, we can use the Rational Root Theorem to test possible rational roots. After testing a few values, we find that \( x = -2 \) is a root. We can factor the polynomial by using synthetic division or polynomial long division, yielding: \[(x + 2)(x^2 - 2x + 6) = 0.\] Next, we can find the coordinates of the solutions by solving for \( x \) in: 1. \( x = -2 \) gives \( y = -92(-2) + 10 = 184 + 10 = 194 \). For \( x^2 - 2x + 6 = 0 \), we find the discriminant: \[ (-2)^2 - 4(1)(6) = 4 - 24 = -20. \] Since the discriminant is negative, this part has no real solutions. Therefore, the only solution in real coordinates is: \[ (-2, 194). \] So the coordinates in exact form are \( \boxed{(-2, 194)}. \)