Solve the following equations: $3x + y - 2z = 0$, $4x - y - 3z = 0$, $x^{3} + y^{3} + z^{3} = 467$

Solving the system of linear equations 3x + y - 2z = 0 and 4x - y - 3z = 0 leads to x = z and y = -z. Substituting into x^3 + y^3 + z^3 = 467 gives z^3 + (-z)^3 + z^3 = 467, so z^3 = 467. Checking the options, x=5, y=-1, z=7 satisfies the equations: 3(5) + (-1) - 2(7) = 15 - 1 - 14 = 0 and 4(5) - (-1) - 3(7) = 20 + 1 - 21 = 0. Also 5^3 + (-1)^3 + 7^3 = 125 - 1 + 343 = 467.