To determine the number of solutions for the system of equations using the substitution method, we will substitute \( y \) from the linear equation into the quadratic equation.
The given equations are:
- \( y = x^2 - 9x + 39 \)
- \( y = 9x - 42 \)
First, we substitute \( y \) from the second equation into the first equation:
\[ 9x - 42 = x^2 - 9x + 39 \]
Next, we rearrange this equation to set it to zero:
\[ 0 = x^2 - 9x + 39 - 9x + 42 \]
Combine like terms:
\[ 0 = x^2 - 18x + 81 \]
Now we can rewrite this as:
\[ x^2 - 18x + 81 = 0 \]
Next, we will factor the quadratic equation:
\[ (x - 9)(x - 9) = 0 \]
This can be rewritten as:
\[ (x - 9)^2 = 0 \]
This equation has a double root at \( x = 9 \).
Now we can find the corresponding value of \( y \) by substituting \( x = 9 \) back into one of the original equations. We will use the linear equation:
\[ y = 9(9) - 42 = 81 - 42 = 39 \]
So, the point of intersection (or the solution to the system) is \( (9, 39) \).
This system has exactly 1 solution which occurs at the point \( (9, 39) \). Thus, the answer is that this system has 1 solution.