Answer:
The solutions for the given system of equations are:
[tex]\left \{ {{x=-5} \atop {y=12-4z}} \right.[/tex]
Step-by-step explanation:
Given the equation system:
[tex]\left \{ {{3x+y+4z=-3} \atop {-x+y+4z=17}} \right.[/tex]
We obtain the following matrix:
[tex]\left[\begin{array}{cccc}3&1&4&-3\\-1&1&4&17\end{array}\right][/tex]
Step 1: Multiply the fisrt row by 1/3.
[tex]\left[\begin{array}{cccc}1&\frac{1}{3} &\frac{4}{3}&-1\\-1&1&4&17\end{array}\right][/tex]
Step 2: Sum the first row and the second row.
[tex]\left[\begin{array}{cccc}1&\frac{1}{3} &\frac{4}{3}&-1\\0&\frac{4}{3} &\frac{16}{3}&16\end{array}\right][/tex]
Step 3: Multiply the second row by 3/4.
[tex]\left[\begin{array}{cccc}1&\frac{1}{3} &\frac{4}{3}&-1\\0&1 &4&12\end{array}\right][/tex]
Step 4: Multiply the second row by -1/3 and sum the the first row.
[tex]\left[\begin{array}{cccc}1&0 &0&-5\\0&1 &4&12\end{array}\right][/tex]
The result of the reduced matrix is:
[tex]\left \{ {{x=-5} \atop {y+4z=12}} \right.[/tex]
This is equal to:
[tex]\left \{ {{x=-5} \atop {y=12-4z}} \right.[/tex]
These are the solutions for the system of equations in terms of z, where z can be any number.
