To determine the number of solutions for the given system of equations:
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Identifying the equations: \[ \text{Equation 1: } 2x + y = 6 \] \[ \text{Equation 2: } 4x + 2y = 8 \]
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Simplifying the second equation: The second equation can be rewritten to see if it's a multiple of the first equation: \[ 4x + 2y = 8 \quad \text{(divide by 2)} \] \[ 2x + y = 4 \]
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Now we have two equations:
- From the first equation: \( 2x + y = 6 \)
- From the simplified second equation: \( 2x + y = 4 \)
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Comparing the two equations: The left-hand sides of both equations are the same (\(2x + y\)), but the right-hand sides are different: \[ 6 \neq 4 \] This indicates that the two equations are inconsistent.
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Conclusion: Since the two equations represent parallel lines that never intersect, the system has no solutions.
The number of solutions to the system of equations is 0.