Final answer:
The system of equations 2x + y = 7 and -6x = 3y - 21 represents the same line and therefore has infinitely many solutions.
Step-by-step explanation:
To determine if the system of equations 2x + y = 7 and -6x = 3y - 21 has one, many, or no solutions, we will use the method of substitution or elimination. First, let us rearrange the second equation:
-6x = 3y - 21
Divide both sides by -3 to simplify:
2x = -y + 7
Now, we observe that this equation is essentially the same as the first equation after being multiplied by -1. This means that both equations represent the same line. Therefore, the system has infinitely many solutions, as any point that lies on this line will satisfy both equations.