To determine if the point \((1, 6)\) is a solution to the given system of equations, we need to substitute \(x = 1\) and \(y = 6\) into both equations.
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For the first equation: \[ -x + y = 5 \] Substituting \(x = 1\) and \(y = 6\): \[ -1 + 6 = 5 \] This simplifies to: \[ 5 = 5 \] This equation is satisfied.
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For the second equation: \[ y = (x - 3)^2 + 2 \] Substituting \(x = 1\) and \(y = 6\): \[ 6 = (1 - 3)^2 + 2 \] Calculate \((1 - 3)^2\): \[ (1 - 3)^2 = (-2)^2 = 4 \] Now substitute back into the equation: \[ 6 = 4 + 2 \] This simplifies to: \[ 6 = 6 \] This equation is also satisfied.
Since both equations are satisfied with the point \((1, 6)\), we conclude that \((1, 6)\) is indeed a solution to the system of equations.