To determine which of the given systems of linear equations has no solution, we should identify cases where the lines represented by the equations are parallel. Two lines are parallel if they have the same slope but different y-intercepts.
Let's analyze each system:
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System 1: \[ \begin{align*} 4x + y &= 17 \quad (1) \ 4x - y &= 3 \quad (2) \end{align*} \]
For (1), rearranging gives \( y = -4x + 17 \). For (2), rearranging gives \( -y = -4x + 3 \) or \( y = 4x - 3 \). Since the slopes are different (1 for the first equation and -4 for the second), this system has a solution.
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System 2: \[ \begin{align*} 4x - y &= -17 \quad (1) \ 4x + y &= 3 \quad (2) \end{align*} \]
For (1), rearranging gives \( y = 4x + 17 \). For (2), rearranging gives \( y = -4x + 3 \). Again, the slopes are different, so this system has a solution.
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System 3: \[ \begin{align*} 4x - y &= 17 \quad (1) \ 4x + y &= 3 \quad (2) \end{align*} \]
For (1), rearranging gives \( y = 4x - 17 \). For (2), rearranging gives \( y = -4x + 3 \). The slopes are different, so this system has a solution.
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System 4: \[ \begin{align*} 4x - y &= 17 \quad (1) \ 4x - y &= 3 \quad (2) \end{align*} \]
Both equations are essentially the same type of line but yield different constant terms. Therefore, they are parallel and there is no solution.
Thus, the system of equations that has no solution is:
{4x−y=17, 4x−y=3}.
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