To determine which of the given systems describes a zero-solution (i.e., a system with no solutions, where the lines are parallel), we can analyze each pair of equations. A system has no solutions if the lines are parallel, which occurs when they have the same slope but different y-intercepts.
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System 1: \( y = -12x + 2 \) and \( 3x + 6y = 5 \)
Rearranging the second equation: \[ 6y = -3x + 5 \quad \Rightarrow \quad y = -\frac{1}{2}x + \frac{5}{6} \] The slopes are -12 and -1/2. Not parallel.
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System 2: \( y = 12x + 6 \) and \( -6 - 2y = x \)
Rearranging the second equation: \[ -2y = x + 6 \quad \Rightarrow \quad 2y = -x - 6 \quad \Rightarrow \quad y = -\frac{1}{2}x - 3 \] The slopes are 12 and -1/2. Not parallel.
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System 3: \( y = x \) and \( y = -x \)
The slopes are 1 and -1. Not parallel.
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System 4: \( 2y + x = 3 \) and \( 4x + 8y = 12 \)
Rearranging the first equation: \[ 2y = -x + 3 \quad \Rightarrow \quad y = -\frac{1}{2}x + \frac{3}{2} \] The second equation can be simplified: \[ 4x + 8y = 12 \quad \Rightarrow \quad y = -\frac{1}{2}x + \frac{3}{2} \] The slopes are both -1/2, and since they have the same y-intercept, they are actually the same line. Not zero-solution.
After analyzing all options, it appears that none of these systems have a zero-solution. If we were to choose the answer that is closest to a zero-solution in this context, we would need to identify a pair of equations with the same slope but different y-intercepts—which none of these fulfill. Therefore, none of the systems listed describes a zero-solution (no solutions).
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