Final answer:
The first system of linear equations has a unique solution, while the number of solutions for the second system is indeterminate without further information.
Step-by-step explanation:
In the first system of linear equations, -2x + 3y = 9 and -4x + 12y = 12, we can determine the number of solutions without solving the system by comparing the slopes of the two equations. If the slopes are equal, then the system has infinitely many solutions. If the slopes are not equal but the y-intercepts are, then the system has no solution. If the slopes and y-intercepts are not equal, then the system has a unique solution. In this case, the slopes are not equal, so we can conclude that the system has a unique solution.
In the second system of linear equations, 2x - 3y - 4z = 0 and 6x = 3y + 12, we can also determine the number of solutions without solving the system. Here, the first equation is a linear equation in three variables and the second equation is a linear equation in two variables. Since the number of variables is different, these equations represent planes in three-dimensional space. Therefore, they can have a unique solution (intersection of the two planes), infinitely many solutions (the two planes are the same), or no solution (the two planes are parallel and do not intersect). In this case, we cannot determine the number of solutions without further information, so the answer is indeterminate.