To determine the number of solutions for the system of equations
1. \( y = x^2 - 9x + 39 \)
2. \( y = 9x - 42 \)
we can use the substitution method. Since both equations equal \( y \), we can set them equal to each other:
\[
x^2 - 9x + 39 = 9x - 42
\]
Now, we'll rearrange this equation to form a standard quadratic equation:
\[
x^2 - 9x - 9x + 39 + 42 = 0
\]
This simplifies to:
\[
x^2 - 18x + 81 = 0
\]
Next, we can factor this quadratic equation:
\[
(x - 9)(x - 9) = 0
\]
This gives us:
\[
(x - 9)^2 = 0
\]
The solution to this equation is:
\[
x - 9 = 0 \implies x = 9
\]
Now, substituting \( x = 9 \) back into one of the original equations to find \( y \):
Using \( y = 9x - 42 \):
\[
y = 9(9) - 42 = 81 - 42 = 39
\]
So the solution to the system is \( (9, 39) \).
Since we arrived at a single value of \( x \) which gives a corresponding value of \( y \), the system has **1 solution**.
Thus, the answer is:
This system has **1 solution**.