<s>[INST] How to solve this system without WolframAlpha How can I solve this system without using WolframAlpha or any other program?
$$\begin{equation}
   \begin{cases}
   2x_1+\lambda x_2+3x_3+4x_4=1\\x_1-x_2+9x_3+7x_4=3\\3x_1+5x_2+\lambda x_3+5x_4=1\\x_1+2x_2+x_4=0
   \end{cases}
\end{equation}$$
 [/INST] i will use $k$ for $\lambda.$ look at the augmented matrix 
$\begin{align}\pmatrix{2&k&3&4&1&|a\\1&-1&9&7&3&|b\\3&5&k&5&1&|c\\1&2&0&1&1&|d} 
&\to  \pmatrix{1&2&0&1&1&|d\\1&-1&9&7&3&|b\\3&5&k&5&1&|c\\2&k&3&4&1&|a} \\
&\to \pmatrix{1&2&0&1&1&|d\\0&-3&9&6&-2&|b-d\\0&-1&k&3&-2&|c-3d\\0&k-4&3&2&-1&|a-2d} \\
&\to \pmatrix{1&2&0&1&1&|d\\0&1&-k&-3&2&|-c+3d\\0&0&9-3k&0&4&|b-d+3(-c+3d)\\
0&0&3-k(4-k)&2-3(4-k)&-1+2(4-k)&|a-2d+(4-k)(-c+3d)} 
\end{align}$
now you have to look at different cases. for example if $k = 3.$
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