To determine which system of equations describes a zero-solution (where the system has no solution), we need to find systems of equations that represent parallel lines. Parallel lines do not intersect, which means they have no points in common, leading to a zero-solution.
Let's analyze the provided options:
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Option A:
- \(2y + x = 3\)
- \(4x + 8y = 12\)
Let's manipulate the second equation:
- \(4x + 8y = 12\) can be simplified to \(x + 2y = 3\) (dividing the whole equation by 4).
- This simplifies to the same equation as the first one since \(2y + x = 3\) and \(x + 2y = 3\) are equivalent. This system has infinitely many solutions (the same line).
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Option B:
- \(6y = -12x + 18\)
- \(8x + 4y = 2\)
Let's manipulate these equations.
- The first equation can be rewritten as \(y = -2x + 3\).
- The second equation can be rewritten as \(4y = -8x + 2\) or \(y = -2x + \frac{1}{2}\).
- Since their slopes are the same (\(-2\)) but their y-intercepts are different (3 vs. \( \frac{1}{2} \)), these lines are parallel and do not intersect. They have no solutions.
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Option C:
- \(y = x\)
- \(y = -x\)
These lines intersect at the origin (0,0). Thus, they have a solution.
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Option D:
- \(y = \frac{1}{2}x + 6\)
- \(-6 - 2y = x\) can be rearranged to \(y = -\frac{1}{2}x - 3\).
Here, the slopes are \(\frac{1}{2}\) and \(-\frac{1}{2}\), so these lines are not parallel and will intersect at some point, meaning they have a solution.
Based on this analysis, the correct answer is Option B:
- \(6y = -12x + 18\) and \(8x + 4y = 2\) describe a system with no solutions (i.e., a zero-solution).