To determine the number of real solutions for the system of equations given by \( y = -2x + 1 \) (a linear equation) and \( y = -x^2 - 1 \) (a quadratic equation), we can set the two equations equal to each other:
\[ -2x + 1 = -x^2 - 1. \]
Rearranging this equation, we get:
\[ x^2 - 2x + 2 = 0. \]
Next, we can use the quadratic formula to find the solutions for \( x \):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] where \( a = 1 \), \( b = -2 \), and \( c = 2 \).
Calculating the discriminant (\( b^2 - 4ac \)):
\[ (-2)^2 - 4(1)(2) = 4 - 8 = -4. \]
Since the discriminant is negative, this means that the quadratic equation has no real solutions.
Thus, the best characterization of the number of real solutions for the system of equations is:
The system of equations has no real solutions.
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