To determine which system of equations has exactly one solution, we need to analyze the relationships between the equations in each pair.
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\(y = -8x - 6\) and \(y = -8x + 6\):
These two equations have the same slope (-8), which means they are parallel lines. Since parallel lines never intersect, this system has no solutions. -
\(y = -8x - 6\) and \( \frac{1}{2}y = -4x - 3\):
First, we can rewrite the second equation in terms of y: \[ \frac{1}{2}y = -4x - 3 \implies y = -8x - 6 \] This means both equations are actually the same line. Hence, there are infinitely many solutions. -
\(y = -8x - 6\) and \(y = 8x - 6\):
In this case, the slopes are -8 and 8, so these lines are not parallel and will intersect at exactly one point. This system has exactly one solution. -
\(y = -8x - 6\) and \(-y = 8x + 6\):
Rewriting the second equation gives: \[ y = -8x - 6 \] Again, both equations are the same line, leading to infinitely many solutions.
From the analysis, the system that has exactly one solution is:
\(y = -8x - 6\) and \(y = 8x - 6\).