To determine which system of linear equations has no solution, we can analyze each option for the possibility of having parallel lines, which implies that they will never intersect.
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System 1: \[ \begin{align*} 4x - y &= 17 \ 4x - y &= 3 \end{align*} \] Here, both equations have the same left-hand side but different right-hand sides, indicating that they represent parallel lines. Thus, this system has no solution.
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System 2: \[ \begin{align*} 4x - y &= -17 \ 4x + y &= 3 \end{align*} \] To determine if this has a solution, we can solve the second equation for \( y \): \( y = 3 - 4x \). Substituting this into the first equation does not yield a contradiction, so this system has a solution.
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System 3: \[ \begin{align*} 4x - y &= 17 \ 4x + y &= 3 \end{align*} \] Again, we can solve the second equation for \( y \): \( y = 3 - 4x \). Substituting into the first equation shows that this system also has a solution.
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System 4: \[ \begin{align*} 4x + y &= 17 \ 4x - y &= 3 \end{align*} \] Here, solving the second equation for \( y \) gives \( y = 4x - 3 \). As before, substituting this into the first equation indicates that this system also has a solution.
Conclusion: The only system with no solution is the first one: \[ {4x−y=17, , 4x−y=3} \]
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