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(C) The equation of a plane passing through three points $(x_1, y_1, z_1)$,$(x_2, y_2, z_2)$,and $(x_3, y_3, z_3)$ is given by the determinant equatio

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$5x + 6y + 2z - 23 = 0$

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(C) The equation of a plane passing through three points $(x_1, y_1, z_1)$,$(x_2, y_2, z_2)$,and $(x_3, y_3, z_3)$ is given by the determinant equation: $\left|\begin{array}{ccc} x-x_1 & y-y_1 & z-z_1 \ x_2-x_1 & y_2-y_1 & z_2-z_1 \ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{array}ight| = 0$ Substituting the given points $(3,1,1)$,$(1,2,3)$,and $(-1,4,2)$: $\left|\begin{array}{ccc} x-3 & y-1 & z-1 \ 1-3 & 2-1 & 3-1 \ -1-3 & 4-1 & 2-1 \end{array}ight| = 0$ $\Rightarrow \left|\begin{array}{ccc} x-3 & y-1 & z-1 \ -2 & 1 & 2 \ -4 & 3 & 1 \end{array}ight| = 0$ Expanding along the first row: $(x-3)(1-6) - (y-1)(-2+8) + (z-1)(-6+4) = 0$ $-5(x-3) - 6(y-1) - 2(z-1) = 0$ $-5x + 15 - 6y + 6 - 2z + 2 = 0$ $-5x - 6y - 2z + 23 = 0$ Multiplying by $-1$: $5x + 6y + 2z - 23 = 0$

The Cartesian equation of the plane,passing through the points $(3,1,1)$,$(1,2,3)$ and $(-1,4,2)$,is

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