To determine which of the given systems describes a zero-solution, we need to identify systems of equations that are inconsistent, meaning they have no solution at all, rather than a solution of zero.
Let's analyze each option:
-
y = -1/2x + 2 and 3x + 6y = 5
- The first equation is a line with slope -1/2. The second can be rewritten as y = -1/2x + 5/6. These are parallel lines because they have the same slope (-1/2) but different y-intercepts (2 vs 5/6). Therefore, they do not intersect, and this system has no solutions.
-
y = 1/2x + 6 and -6 - 2y = x
- The second equation can be rewritten as 2y = -x - 6 or y = -1/2x - 3. These two equations have different slopes (1/2 vs -1/2) and will intersect at one point, indicating that this system has a unique solution.
-
y = x and y = -x
- These two lines are perpendicular and intersect at the origin (0, 0). This system has one solution, specifically (0, 0).
-
2y + x = 3 and 4x + 8y = 12
- The second equation can be simplified as 2y + x = 3 (by dividing the entire equation by 4). This means both equations are actually the same line, which means there are infinitely many solutions, not zero.
From the above analysis, the first option:
y = -1/2x + 2 and 3x + 6y = 5
is the only system that describes a zero-solution, as it is inconsistent (the lines are parallel).