To determine which system of linear equations has no solution, we need to look for a scenario where the lines represented by the equations are parallel (i.e., they have the same slope but different intercepts).
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First system: \[ \begin{align*} 4x - y &= -17 \quad \text{(Equation 1)} \ 4x + y &= 3 \quad \text{(Equation 2)} \end{align*} \]
- Rearranged:
- \(y = 4x + 17\) (slope = 4)
- \(y = -4x + 3\) (slope = -4)
- These lines intersect (different slopes).
- Rearranged:
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Second system: \[ \begin{align*} 4x + y &= 17 \quad \text{(Equation 1)} \ 4x - y &= 3 \quad \text{(Equation 2)} \end{align*} \]
- Rearranged:
- \(y = -4x + 17\) (slope = -4)
- \(y = 4x - 3\) (slope = 4)
- These lines intersect (different slopes).
- Rearranged:
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Third system: \[ \begin{align*} 4x - y &= 17 \quad \text{(Equation 1)} \ 4x + y &= 3 \quad \text{(Equation 2)} \end{align*} \]
- Rearranged:
- \(y = 4x - 17\) (slope = 4)
- \(y = -4x + 3\) (slope = -4)
- These lines intersect (different slopes).
- Rearranged:
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Fourth system: \[ \begin{align*} 4x - y &= 17 \quad \text{(Equation 1)} \ 4x - y &= 3 \quad \text{(Equation 2)} \end{align*} \]
- Both equations \(4x - y\) represent the same line if simplified. However, since they equal different constants, this results in contradictions (i.e., a parallel line).
Now, out of these analyses, the fourth system has no solution because it represents two identical lines with different constant values.
Thus, the answer is: \[ {4x - y = 17, , 4x - y = 3} \] which has no solution.
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