To determine which system of equations describes a zero-solution (meaning that the system has no solution), we will analyze each system for inconsistency.
- System 1: \[ 2y + x = 3 \] \[ 4x + 8y = 12 \] The second equation can be simplified by dividing everything by 4: \[ x + 2y = 3 \] This is equivalent to the first equation, which means these two lines are the same. Therefore, there are infinitely many solutions (not zero).
- System 2: \[ 6y = -12x + 18 \quad(1) \] \[ 8x + 4y = 2 \quad(2) \] Rewriting the first equation gives: \[ y = -2x + 3 \] For the second equation, solving for y: \[ 4y = 2 - 8x \implies y = \frac{1}{2} - 2x \] Now we set the two expressions for y equal to find any possible solutions: \[ -2x + 3 = \frac{1}{2} - 2x \] After simplifying, both equations lead to a false statement (such as 3 = 0), indicating that this system has no solutions (zero-solution).
- System 3: \[ y = x \] \[ y = -x \] These two equations are lines that intersect at the origin (0,0). Therefore, there is one solution (not zero).
- System 4: \[ y = 12x + 6 \] \[ -6 - 2y = x \implies 2y = -x - 6 \implies y = -\frac{1}{2}x - 3 \] These two lines have different slopes (one has a slope of 12, the other has a slope of -1/2) and will intersect at some point. Thus, there is one solution (not zero).
After analyzing all the systems, System 2 (6y = -12x + 18 and 8x + 4y = 2) is the only one that yields a zero-solution (no solution).
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