tell wether the systems has one solutin, infinitively many, or no solution.
1.x-3y=27; 2x=6y-14
2.3x-5y=-2; x=3y=4
3.x+2y=6; 2x-4y=-12
4.5x+y=15; 3y=-15x+6
5.3x=4y-5; 12y=9x+15
6.3x-y=-2; -2x+2y=8

Question

tell wether the systems has one solutin, infinitively many, or no solution. 
1.x-3y=27; 2x=6y-14
2.3x-5y=-2; x=3y=4
3.x+2y=6; 2x-4y=-12
4.5x+y=15; 3y=-15x+6
5.3x=4y-5; 12y=9x+15
6.3x-y=-2; -2x+2y=8

dape · Answer

It has one solution if the lines the equations describe are not parallel. It has no solutions if they are parallel and does not coincide, if they coincide it has infinitely many solutions.

dape · Answer

To solve the problem you can just try and solve each system. A solid method to do this is to solve the first equation after either x or y and then replace all x's or y's in the second equation by this solution and solve the resulting equation. I will solve the first problem as an example.

dape · Answer

$$ x-3y=27 \ x=27+3y $$
putting this into the second equation gives
$$ 2(27+3y)=6y-14  \
54+6y=6y-14 $$
We see that this never holds. So it has no solution, that means that the corresponding lines are parallel and never intersect.