To determine the number of solutions for the system of equations using the substitution method, we start by substituting one equation into the other.
We have:
- \( y = x^2 - 2x - 6 \) (Equation 1)
- \( y = -5x - 9 \) (Equation 2)
We can substitute Equation 2 into Equation 1. This means we will replace \( y \) in Equation 1 with the expression from Equation 2:
\[ -5x - 9 = x^2 - 2x - 6 \]
Next, we'll rearrange the equation by moving all terms to one side:
\[ 0 = x^2 - 2x - 6 + 5x + 9 \]
Combining like terms gives:
\[ 0 = x^2 + 3x + 3 \]
Now, we need to determine the number of solutions to the quadratic equation \( x^2 + 3x + 3 = 0 \). We can do this by calculating the discriminant (\( D \)):
\[ D = b^2 - 4ac \] where \( a = 1 \), \( b = 3 \), and \( c = 3 \).
\[ D = 3^2 - 4(1)(3) = 9 - 12 = -3 \]
Since the discriminant \( D \) is negative (\( D < 0 \)), this indicates that the quadratic equation has no real solutions.
Thus, the system of equations has 0 solutions.