To determine the number of solutions for the given system of equations using the elimination method, we will rewrite the equations and analyze them.
The given equations are:
- \( y = x^2 - 4x + 10 \) (a quadratic equation)
- \( y = -2x + 5 \) (a linear equation)
To find the intersection points, we can set the two equations equal to each other: \[ x^2 - 4x + 10 = -2x + 5 \]
Now, let's rearrange the equation: \[ x^2 - 4x + 2x + 10 - 5 = 0 \] \[ x^2 - 2x + 5 = 0 \]
Now we will use the quadratic formula to solve for \(x\): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 1\), \(b = -2\), and \(c = 5\).
Calculating the discriminant \(b^2 - 4ac\): \[ (-2)^2 - 4(1)(5) = 4 - 20 = -16 \]
Since the discriminant is negative (-16), this indicates that there are no real solutions to the equation. This means that the quadratic and linear equations do not intersect.
Thus, the system has 0 solutions.
The correct answer is: 0