Solve {l}{kx_{1}+kx_{2}+4x_{3}=2}{(k+1)x_{1}+(k+3)x_{2}-kx_{3}=2}{(k+2)x_{1}+(k-2)x_{2}-(k-1)x_{3}=h+2}

$$\left\{ \begin{array} { l } { k x _ { 1 } + k x _ { 2 } + 4 x _ { 3 } = 2 } \\ { ( k + 1 ) x _ { 1 } + ( k + 3 ) x _ { 2 } - k x _ { 3 } = 2 } \\ { ( k + 2 ) x _ { 1 } + ( k - 2 ) x _ { 2 } - ( k - 1 ) x _ { 3 } = h + 2 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=\frac{hk^{2}+4hk+12h+10k+34}{2\left(3k^{2}+11k+16\right)}$$

$$x_{2}=-\frac{hk^{2}+4hk+4h-2k-10}{2\left(3k^{2}+11k+16\right)}$$

$$x_{3}=\frac{8-hk}{3k^{2}+11k+16}$$

Short Steps Using Substitution

Solve kx_{1}+kx_{2}+4x_{3}=2 for x_{3}.

$$x_{3}=-\frac{1}{4}kx_{1}-\frac{1}{4}kx_{2}+\frac{1}{2}$$

Substitute -\frac{1}{4}kx_{1}-\frac{1}{4}kx_{2}+\frac{1}{2} for x_{3} in the second and third equation.

$$\left(k+1\right)x_{1}+\left(k+3\right)x_{2}-k\left(-\frac{1}{4}kx_{1}-\frac{1}{4}kx_{2}+\frac{1}{2}\right)=2 \left(k+2\right)x_{1}+\left(k-2\right)x_{2}-\left(k-1\right)\left(-\frac{1}{4}kx_{1}-\frac{1}{4}kx_{2}+\frac{1}{2}\right)=h+2$$

Solve these equations for x_{2} and x_{1} respectively.

$$x_{2}=8\left(12+4k+k^{2}\right)^{-1}+2\left(12+4k+k^{2}\right)^{-1}k-4\left(12+4k+k^{2}\right)^{-1}x_{1}-4\left(12+4k+k^{2}\right)^{-1}kx_{1}-\left(12+4k+k^{2}\right)^{-1}k^{2}x_{1} x_{1}=6\left(8+3k+k^{2}\right)^{-1}+2\left(8+3k+k^{2}\right)^{-1}k+4\left(8+3k+k^{2}\right)^{-1}h+8\left(8+3k+k^{2}\right)^{-1}x_{2}-3\left(8+3k+k^{2}\right)^{-1}kx_{2}-\left(8+3k+k^{2}\right)^{-1}k^{2}x_{2}$$

Substitute 8\left(12+4k+k^{2}\right)^{-1}+2\left(12+4k+k^{2}\right)^{-1}k-4\left(12+4k+k^{2}\right)^{-1}x_{1}-4\left(12+4k+k^{2}\right)^{-1}kx_{1}-\left(12+4k+k^{2}\right)^{-1}k^{2}x_{1} for x_{2} in the equation x_{1}=6\left(8+3k+k^{2}\right)^{-1}+2\left(8+3k+k^{2}\right)^{-1}k+4\left(8+3k+k^{2}\right)^{-1}h+8\left(8+3k+k^{2}\right)^{-1}x_{2}-3\left(8+3k+k^{2}\right)^{-1}kx_{2}-\left(8+3k+k^{2}\right)^{-1}k^{2}x_{2}.

$$x_{1}=6\left(8+3k+k^{2}\right)^{-1}+2\left(8+3k+k^{2}\right)^{-1}k+4\left(8+3k+k^{2}\right)^{-1}h+8\left(8+3k+k^{2}\right)^{-1}\left(8\left(12+4k+k^{2}\right)^{-1}+2\left(12+4k+k^{2}\right)^{-1}k-4\left(12+4k+k^{2}\right)^{-1}x_{1}-4\left(12+4k+k^{2}\right)^{-1}kx_{1}-\left(12+4k+k^{2}\right)^{-1}k^{2}x_{1}\right)-3\left(8+3k+k^{2}\right)^{-1}k\left(8\left(12+4k+k^{2}\right)^{-1}+2\left(12+4k+k^{2}\right)^{-1}k-4\left(12+4k+k^{2}\right)^{-1}x_{1}-4\left(12+4k+k^{2}\right)^{-1}kx_{1}-\left(12+4k+k^{2}\right)^{-1}k^{2}x_{1}\right)-\left(8+3k+k^{2}\right)^{-1}k^{2}\left(8\left(12+4k+k^{2}\right)^{-1}+2\left(12+4k+k^{2}\right)^{-1}k-4\left(12+4k+k^{2}\right)^{-1}x_{1}-4\left(12+4k+k^{2}\right)^{-1}kx_{1}-\left(12+4k+k^{2}\right)^{-1}k^{2}x_{1}\right)$$

Solve x_{1}=6\left(8+3k+k^{2}\right)^{-1}+2\left(8+3k+k^{2}\right)^{-1}k+4\left(8+3k+k^{2}\right)^{-1}h+8\left(8+3k+k^{2}\right)^{-1}\left(8\left(12+4k+k^{2}\right)^{-1}+2\left(12+4k+k^{2}\right)^{-1}k-4\left(12+4k+k^{2}\right)^{-1}x_{1}-4\left(12+4k+k^{2}\right)^{-1}kx_{1}-\left(12+4k+k^{2}\right)^{-1}k^{2}x_{1}\right)-3\left(8+3k+k^{2}\right)^{-1}k\left(8\left(12+4k+k^{2}\right)^{-1}+2\left(12+4k+k^{2}\right)^{-1}k-4\left(12+4k+k^{2}\right)^{-1}x_{1}-4\left(12+4k+k^{2}\right)^{-1}kx_{1}-\left(12+4k+k^{2}\right)^{-1}k^{2}x_{1}\right)-\left(8+3k+k^{2}\right)^{-1}k^{2}\left(8\left(12+4k+k^{2}\right)^{-1}+2\left(12+4k+k^{2}\right)^{-1}k-4\left(12+4k+k^{2}\right)^{-1}x_{1}-4\left(12+4k+k^{2}\right)^{-1}kx_{1}-\left(12+4k+k^{2}\right)^{-1}k^{2}x_{1}\right) for x_{1}.

$$x_{1}=17\left(16+11k+3k^{2}\right)^{-1}+5\left(16+11k+3k^{2}\right)^{-1}k+6\left(16+11k+3k^{2}\right)^{-1}h+2\left(16+11k+3k^{2}\right)^{-1}kh+\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}k^{2}h$$

Substitute 17\left(16+11k+3k^{2}\right)^{-1}+5\left(16+11k+3k^{2}\right)^{-1}k+6\left(16+11k+3k^{2}\right)^{-1}h+2\left(16+11k+3k^{2}\right)^{-1}kh+\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}k^{2}h for x_{1} in the equation x_{2}=8\left(12+4k+k^{2}\right)^{-1}+2\left(12+4k+k^{2}\right)^{-1}k-4\left(12+4k+k^{2}\right)^{-1}x_{1}-4\left(12+4k+k^{2}\right)^{-1}kx_{1}-\left(12+4k+k^{2}\right)^{-1}k^{2}x_{1}.

$$x_{2}=8\left(12+4k+k^{2}\right)^{-1}+2\left(12+4k+k^{2}\right)^{-1}k-4\left(12+4k+k^{2}\right)^{-1}\left(17\left(16+11k+3k^{2}\right)^{-1}+5\left(16+11k+3k^{2}\right)^{-1}k+6\left(16+11k+3k^{2}\right)^{-1}h+2\left(16+11k+3k^{2}\right)^{-1}kh+\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}k^{2}h\right)-4\left(12+4k+k^{2}\right)^{-1}k\left(17\left(16+11k+3k^{2}\right)^{-1}+5\left(16+11k+3k^{2}\right)^{-1}k+6\left(16+11k+3k^{2}\right)^{-1}h+2\left(16+11k+3k^{2}\right)^{-1}kh+\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}k^{2}h\right)-\left(12+4k+k^{2}\right)^{-1}k^{2}\left(17\left(16+11k+3k^{2}\right)^{-1}+5\left(16+11k+3k^{2}\right)^{-1}k+6\left(16+11k+3k^{2}\right)^{-1}h+2\left(16+11k+3k^{2}\right)^{-1}kh+\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}k^{2}h\right)$$

Calculate x_{2} from x_{2}=8\left(12+4k+k^{2}\right)^{-1}+2\left(12+4k+k^{2}\right)^{-1}k-4\left(12+4k+k^{2}\right)^{-1}\left(17\left(16+11k+3k^{2}\right)^{-1}+5\left(16+11k+3k^{2}\right)^{-1}k+6\left(16+11k+3k^{2}\right)^{-1}h+2\left(16+11k+3k^{2}\right)^{-1}kh+\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}k^{2}h\right)-4\left(12+4k+k^{2}\right)^{-1}k\left(17\left(16+11k+3k^{2}\right)^{-1}+5\left(16+11k+3k^{2}\right)^{-1}k+6\left(16+11k+3k^{2}\right)^{-1}h+2\left(16+11k+3k^{2}\right)^{-1}kh+\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}k^{2}h\right)-\left(12+4k+k^{2}\right)^{-1}k^{2}\left(17\left(16+11k+3k^{2}\right)^{-1}+5\left(16+11k+3k^{2}\right)^{-1}k+6\left(16+11k+3k^{2}\right)^{-1}h+2\left(16+11k+3k^{2}\right)^{-1}kh+\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}k^{2}h\right).

$$x_{2}=-\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}hk^{2}-2\left(16+11k+3k^{2}\right)^{-1}hk+\left(16+11k+3k^{2}\right)^{-1}k-2\left(16+11k+3k^{2}\right)^{-1}h+5\left(16+11k+3k^{2}\right)^{-1}$$

Substitute -\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}hk^{2}-2\left(16+11k+3k^{2}\right)^{-1}hk+\left(16+11k+3k^{2}\right)^{-1}k-2\left(16+11k+3k^{2}\right)^{-1}h+5\left(16+11k+3k^{2}\right)^{-1} for x_{2} and 17\left(16+11k+3k^{2}\right)^{-1}+5\left(16+11k+3k^{2}\right)^{-1}k+6\left(16+11k+3k^{2}\right)^{-1}h+2\left(16+11k+3k^{2}\right)^{-1}kh+\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}k^{2}h for x_{1} in the equation x_{3}=-\frac{1}{4}kx_{1}-\frac{1}{4}kx_{2}+\frac{1}{2}.

$$x_{3}=-\frac{1}{4}k\left(17\left(16+11k+3k^{2}\right)^{-1}+5\left(16+11k+3k^{2}\right)^{-1}k+6\left(16+11k+3k^{2}\right)^{-1}h+2\left(16+11k+3k^{2}\right)^{-1}kh+\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}k^{2}h\right)-\frac{1}{4}k\left(-\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}hk^{2}-2\left(16+11k+3k^{2}\right)^{-1}hk+\left(16+11k+3k^{2}\right)^{-1}k-2\left(16+11k+3k^{2}\right)^{-1}h+5\left(16+11k+3k^{2}\right)^{-1}\right)+\frac{1}{2}$$

Calculate x_{3} from x_{3}=-\frac{1}{4}k\left(17\left(16+11k+3k^{2}\right)^{-1}+5\left(16+11k+3k^{2}\right)^{-1}k+6\left(16+11k+3k^{2}\right)^{-1}h+2\left(16+11k+3k^{2}\right)^{-1}kh+\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}k^{2}h\right)-\frac{1}{4}k\left(-\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}hk^{2}-2\left(16+11k+3k^{2}\right)^{-1}hk+\left(16+11k+3k^{2}\right)^{-1}k-2\left(16+11k+3k^{2}\right)^{-1}h+5\left(16+11k+3k^{2}\right)^{-1}\right)+\frac{1}{2}.

$$x_{3}=-\left(3k^{2}+11k+16\right)^{-1}hk+8\left(3k^{2}+11k+16\right)^{-1}$$

The system is now solved.

$$x_{1}=17\left(16+11k+3k^{2}\right)^{-1}+5\left(16+11k+3k^{2}\right)^{-1}k+6\left(16+11k+3k^{2}\right)^{-1}h+2\left(16+11k+3k^{2}\right)^{-1}kh+\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}k^{2}h x_{2}=-\frac{1}{2}\left(16+11k+3k^{2}\right)^{-1}hk^{2}-2\left(16+11k+3k^{2}\right)^{-1}hk+\left(16+11k+3k^{2}\right)^{-1}k-2\left(16+11k+3k^{2}\right)^{-1}h+5\left(16+11k+3k^{2}\right)^{-1} x_{3}=-\left(3k^{2}+11k+16\right)^{-1}hk+8\left(3k^{2}+11k+16\right)^{-1}$$

Solve for x_1, x_2, x_3 (complex solution)

$$\left\{\begin{matrix}x_{1}=\frac{hk^{2}+4hk+12h+10k+34}{2\left(3k^{2}+11k+16\right)}\text{, }x_{2}=\frac{10+2k-4h-4hk-hk^{2}}{2\left(3k^{2}+11k+16\right)}\text{, }x_{3}=\frac{8-hk}{3k^{2}+11k+16}\text{, }&k\neq \frac{-11+\sqrt{71}i}{6}\text{ and }k\neq \frac{-\sqrt{71}i-11}{6}\\x_{1}=\frac{6-12x_{3}-4kx_{3}-x_{3}k^{2}}{2k}\text{, }x_{2}=\frac{x_{3}k^{2}+4kx_{3}+4x_{3}-2}{2k}\text{, }x_{3}\in \mathrm{C}\text{, }&\left(h=\frac{-11+\sqrt{71}i}{4}\text{ and }k=\frac{-\sqrt{71}i-11}{6}\right)\text{ or }\left(h=\frac{-\sqrt{71}i-11}{4}\text{ and }k=\frac{-11+\sqrt{71}i}{6}\right)\end{matrix}\right.$$