To determine which system of linear equations has no solution, we look for systems that represent parallel lines. Parallel lines have the same slope but different y-intercepts.
Let's analyze each system:
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System:
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\(4x - y = -17\)
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\(4x + y = 3\)
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Convert to slope-intercept form \(y = mx + b\):
- First equation: \(y = 4x + 17\) (slope \(m = 4\))
- Second equation: \(y = -4x + 3\) (slope \(m = -4\))
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Conclusion: Different slopes → has a solution (intersecting lines).
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System:
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\(4x - y = 17\)
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\(4x + y = 3\)
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Convert to slope-intercept form:
- First equation: \(y = 4x - 17\) (slope \(m = 4\))
- Second equation: \(y = -4x + 3\) (slope \(m = -4\))
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Conclusion: Different slopes → has a solution (intersecting lines).
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System:
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\(4x + y = 17\)
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\(4x - y = 3\)
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Convert to slope-intercept form:
- First equation: \(y = -4x + 17\) (slope \(m = -4\))
- Second equation: \(y = 4x - 3\) (slope \(m = 4\))
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Conclusion: Different slopes → has a solution (intersecting lines).
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System:
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\(4x - y = 17\)
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\(4x - y = 3\)
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The first equation can be rewritten as \(y = 4x - 17\) and the second equation as \(y = 4x - 3\). Both equations have the same slope (\(m = 4\)) but different y-intercepts.
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Conclusion: The lines are parallel and never intersect → this system has no solution.
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From the analysis, the system of linear equations that has no solution is:
\[ {4x - y = 17, , 4x - y = 3} \]
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