To find the number of solutions for the given system of equations:
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Write down the equations: \[ 8x - 6y = 48 \quad \text{(1)} \] \[ 8x + 6y = 48 \quad \text{(2)} \]
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Combine the equations: We can add the two equations together to eliminate \(y\): \[ (8x - 6y) + (8x + 6y) = 48 + 48 \] This simplifies to: \[ 16x = 96 \] Dividing both sides by 16, we get: \[ x = 6 \]
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Substitute \(x\) back to find \(y\): Now we can substitute \(x = 6\) back into either of the original equations. Using equation (1): \[ 8(6) - 6y = 48 \] This simplifies to: \[ 48 - 6y = 48 \] Subtracting 48 from both sides gives: \[ -6y = 0 \] Hence, \(y = 0\).
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Conclusion: We found one unique solution: \((x, y) = (6, 0)\). Therefore, the system of equations has one solution.
Answer: one.