1. **State the problem:** We have a system of four linear equations: $$ A(x_0+rV_{01}) + B(y_0+rV_{02}) + C(z_0+rV_{03}) + D = 0 \\ A(x_1+rV_{11}) + B(y_1+rV_{12}) + C(z_1+rV_{13}) + D = 0 \\ A(x_2+rV_{21}) + B(y_2+rV_{22}) + C(z_2+rV_{23}) + D = 0 \\ A(x_3+rV_{31}) + B(y_3+rV_{32}) + C(z_3+rV_{33}) + D = 0 $$ Unknowns are $A, B, C, D$. The goal is to find expressions for these unknowns in terms of $r$ and the given variables. 2. **Rewrite each equation grouping $A, B, C, D$ terms:** $$ A x_i + A r V_{i1} + B y_i + B r V_{i2} + C z_i + C r V_{i3} + D = 0, \quad i=0,1,2,3 $$ Group terms: $$ A(x_i + r V_{i1}) + B(y_i + r V_{i2}) + C(z_i + r V_{i3}) + D = 0 $$ 3. **Matrix form:** Define matrix $M$ and vector $X$: $$ M = \begin{bmatrix} x_0 + r V_{01} & y_0 + r V_{02} & z_0 + r V_{03} & 1 \\ x_1 + r V_{11} & y_1 + r V_{12} & z_1 + r V_{13} & 1 \\ x_2 + r V_{21} & y_2 + r V_{22} & z_2 + r V_{23} & 1 \\ x_3 + r V_{31} & y_3 + r V_{32} & z_3 + r V_{33} & 1 \end{bmatrix}, \quad X = \begin{bmatrix} A \\ B \\ C \\ D \end{bmatrix} $$ The system is: $$ M X = 0 $$ 4. **Nontrivial solution condition:** For $X \neq 0$ to exist, the determinant of $M$ must be zero: $$ \det(M) = 0 $$ This is an equation in $r$. 5. **Solve for $r$:** Calculate $\det(M)$ as a function of $r$ and solve $\det(M) = 0$ for $r$. 6. **Find $A, B, C, D$ for each $r$:** For each root $r_i$, substitute back into $M$ and solve the homogeneous system $M X = 0$. Since the system is homogeneous and singular, solutions form a vector space; find a basis vector for $X$ (up to scale). 7. **Summary:** - Compute $\det(M)$ as a polynomial in $r$. - Solve $\det(M) = 0$ for $r$. - For each $r$, solve $M X = 0$ to find $A, B, C, D$ up to a multiplicative constant. This method finds all $r$ values and corresponding $(A,B,C,D)$ satisfying the system.