To determine if the point \((-2, 6)\) is a solution to the system of equations, we will substitute \(x = -2\) and \(y = 6\) into both equations and see if both equations hold true.
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First equation: \[ x + 2y = 10 \] Substitute \(x = -2\) and \(y = 6\): \[ -2 + 2(6) = -2 + 12 = 10 \] This is true, as the left-hand side equals the right-hand side (10).
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Second equation: \[ 3x + y = 0 \] Substitute \(x = -2\) and \(y = 6\): \[ 3(-2) + 6 = -6 + 6 = 0 \] This is also true, as the left-hand side equals the right-hand side (0).
Since both equations are satisfied when substituting \((-2, 6)\), we can conclude that \((-2, 6)\) is indeed a solution to the system of equations.