Answer:
The system y = 2x + 4 and y = 2x - 12 has no solution because the lines y = 2x + 4 and y = 2x - 12 are parallel and never intersect.
The system -x + 3y = 8 and 2x - 6y = -16 has one solution. To find the solution, we can solve for x or y in one equation and substitute into the other equation. For example, we can solve -x + 3y = 8 for x to get x = 3y - 8, and then substitute into 2x - 6y = -16 to get 2(3y - 8) - 6y = -16. Simplifying this equation gives y = 4, and substituting y = 4 into x = 3y - 8 gives x = 4.
The system x + 2y = 6 and 2x + 4y = 12 has infinitely many solutions. To see why, notice that the second equation is a multiple of the first equation. Specifically, we can multiply the first equation by 2 to get 2x + 4y = 12, which is identical to the second equation. Therefore, any point that satisfies the first equation will also satisfy the second equation. In geometric terms, the two equations represent the same line, so there are infinitely many points that satisfy both equations.