To determine whether the system of equations has one solution, infinite solutions, or no solutions, we can analyze the two equations:
- \( x - 2y = 6 \)
- \( 3x - 6y = 18 \)
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:
- \( x - 2y = 6 \)
- \( x - 2y = 6 \)
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.