does x - 2y = 6, 3x - 6y = 18 have one solution, infinite solutions, or no solutions

To determine whether the system of equations has one solution, infinite solutions, or no solutions, we can analyze the two equations:

1. \( x - 2y = 6 \)
2. \( 3x - 6y = 18 \)

We can simplify the second equation. Notice that the second equation can be expressed as:

\[
3(x - 2y) = 18
\]

Dividing both sides by 3 gives us:

\[
x - 2y = 6
\]

Now we see that both equations are actually the same:

1. \( x - 2y = 6 \)
2. \( x - 2y = 6 \)

Since both equations represent the same line in a coordinate plane, they have infinitely many solutions as every point on this line satisfies both equations.

Thus, the system of equations has **infinite solutions**.