To determine which of the given systems of linear equations has no solution, we need to analyze each pair of equations for consistency. A system of equations has no solution if the lines represented by the equations are parallel, meaning they have the same slope but different y-intercepts.
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System 1: \[ \begin{cases} 4x - y = 17 \ 4x - y = 3 \end{cases} \] These equations represent parallel lines with the same slope (4) but different y-intercepts (17 and 3). This system has no solution.
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System 2: \[ \begin{cases} 4x - y = -17 \ 4x + y = 3 \end{cases} \] These equations have different slopes since the second equation can be rearranged to give \(y = -4x + 3\). They will intersect at one point. This system has a solution.
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System 3: \[ \begin{cases} 4x - y = 17 \ 4x + y = 3 \end{cases} \] Similar to System 2, these equations have different slopes, leading to an intersection point. This one has a solution as well.
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System 4: \[ \begin{cases} 4x + y = 17 \ 4x - y = 3 \end{cases} \] Again, these will produce different slopes, thus they will intersect at a single point. This system also has a solution.
From the analysis above, the only system of linear equations that has no solution is:
{4x−y=17, 4x−y=3}
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