Prove that 4 points lie in the same plane: A (1, 2, -1) B (0, 1, 5) C (-1, 2, 1) D (2, 1, 3).

First, we compose the equation of the plane passing through any three points, for example, through the points A, B and C. The equation of the plane passing through the point A (1; 2; –1) will be: α * (x – 1) + β * ( y – 2) + γ * (z – (- 1)) = 0, where α, β and γ are coefficients that can be determined using the fact that two more points B (0; 1; 5) lie on this plane and C (–1; 2; 1).
The conditions for the passage of the plane through points B and C can be formalized using the following equalities: α * (0 – 1) + β * (1 – 2) + γ * (5 – (- 1)) = 0 and α * (–1 – 1) + β * (2 – 2) + γ * (1 – (- 1)) = 0. Let us simplify these conditions: –α – β + 6 * γ = 0 and – 2 * α + 2 * γ = 0. The last equality allows us to assert that α = γ. Using this equality, we find that β = 5 * γ. Thus, we found out that α: β: γ = 1: 5: 1.
Substituting instead of α, β and γ in the plane equation (see item 1) the proportional numbers 1, 5 and 1, we get 1 * (x – 1) + 5 * (y – 2) + 1 * (z + 1) = 0 or x + 5 * y + z – 10 = 0.
In order to prove that the point D (2; 1; 3) also lies on the plane, it is necessary to substitute the coordinates of this point into the equation of the plane. Then we have 2 + 5 * 1 + 3 – 10 = 2 + 5 + 3 – 10 = 0. This proves that point D also lies on the plane.