To determine which student found the correct solution for the equation \(3x - 4y = 8\), we need to substitute the values of \(x\) and \(y\) from each student’s solution into the equation and see if it holds true.
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Student #1: Linda believes \((1, -4)\) is a solution. \[ 3(1) - 4(-4) = 3 - (-16) = 3 + 16 = 19 \quad \text{(not equal to 8)} \]
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Student #2: Nathan believes \((5, 3)\) is a solution. \[ 3(5) - 4(3) = 15 - 12 = 3 \quad \text{(not equal to 8)} \]
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Student #3: Gary believes \((4, 1)\) is a solution. \[ 3(4) - 4(1) = 12 - 4 = 8 \quad \text{(equal to 8)} \]
Thus, the correct student is Student #3, Gary.
Answer: Student #3 is correct.