A zero-solution means that the system of equations has no solution, which typically occurs when the equations represent parallel lines that never intersect.
Let's analyze each system to determine if they describe a zero-solution:
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\(6y = -12x + 18\) and \(8x + 4y = 2\):
- Rearranging the first equation: \(y = -2x + 3\)
- Rearranging the second equation: \(4y = -8x + 2 \Rightarrow y = -2x + \frac{1}{2}\)
- These lines have the same slope (-2) but different y-intercepts (3 and 0.5), so they are parallel and do not intersect.
- This system describes a zero-solution.
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\(2y + x = 3\) and \(4x + 8y = 12\):
- Rearranging the first equation: \(2y = -x + 3 \Rightarrow y = -\frac{1}{2}x + \frac{3}{2}\)
- Rearranging the second equation: \(8y = -4x + 12 \Rightarrow y = -\frac{1}{2}x + \frac{3}{2}\)
- These lines are actually the same line, so they intersect at infinitely many points.
- This system does not describe a zero-solution.
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\(y = x\) and \(y = -x\):
- These lines intersect at (0,0) and have different slopes (1 and -1).
- This system does not describe a zero-solution.
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\(y = 12x + 6\) and \(-6 - 2y = x\):
- Rearranging the second equation: \(-2y = x + 6 \Rightarrow y = -\frac{1}{2}x - 3\)
- The first equation has a slope of 12 and the second has a slope of -0.5. Since the slopes are different, these lines will intersect at a single point.
- This system does not describe a zero-solution.
Based on this analysis, the only system that describes a zero-solution is:
\(6y = -12x + 18\) and \(8x + 4y = 2\).