91Ó°ÊÓ

The given system of equations is$\left\{\begin{array}{rl}x−2z& =1\\ 2x+3y& =−3\\ 4x−3y−4z& =3\end{array}\right.$

Write the augmented matrix that represents the system of equations

$\left[\begin{array}{rrrr}1& 0& −2& 1\\ 2& 3& 0& −3\\ 4& −3& −4& 3\end{array}\right]$

Perform the operations $R_2=R_2-2R_1$,

$\left[\begin{array}{cccc}1& 0& −2& 1\\ 2−2\left(1\right)& 3−2\left(0\right)& 0−2(−2)& −3−2\left(1\right)\\ 4& −3& −4& 3\end{array}\right]→\left[\begin{array}{rrrr}1& 0& −2& 1\\ 0& 3& 4& −5\\ 4& −3& −4& 3\end{array}\right]$

Now, perform the operations $R_3=R_3-4R_1$,

$\left[\begin{array}{cccc}1& 0& −2& 1\\ 0& 3& 4& −5\\ 4−4\left(1\right)& −3−4\left(0\right)& −4−4(−2)& 3−4\left(1\right)\end{array}\right]→\left[\begin{array}{rrrr}1& 0& −2& 1\\ 0& 3& 4& −5\\ 0& −3& 4& −1\end{array}\right]$

Now, perform the operations $R_2=\frac{R_2}{3}$,

$\left[\begin{array}{rrrr}1& 0& −2& 1\\ \frac{{\displaystyle 0}}{{\displaystyle 3}}& \frac{{\displaystyle 3}}{{\displaystyle 3}}& \frac{{\displaystyle 4}}{{\displaystyle 3}}& −\frac{{\displaystyle 5}}{{\displaystyle 3}}\\ 0& −3& 4& −1\end{array}\right]→\left[\begin{array}{rrrr}1& 0& −2& 1\\ 0& 1& \frac{{\displaystyle 4}}{{\displaystyle 3}}& −\frac{{\displaystyle 5}}{{\displaystyle 3}}\\ 0& −3& 4& −1\end{array}\right]$

Now, perform the operations $R_3=R_3+3R_2$,

$\left[\begin{array}{cccc}1& 0& −2& 1\\ \frac{{\displaystyle 0}}{{\displaystyle 3}}& \frac{{\displaystyle 3}}{{\displaystyle 3}}& \frac{{\displaystyle 4}}{{\displaystyle 3}}& −\frac{{\displaystyle 5}}{{\displaystyle 3}}\\ 0+3\left(\frac{{\displaystyle 0}}{{\displaystyle 3}}\right)& −3+3\left(\frac{{\displaystyle 3}}{{\displaystyle 3}}\right)& 4+3\left(\frac{{\displaystyle 4}}{{\displaystyle 3}}\right)& −1+3(−\frac{{\displaystyle 5}}{{\displaystyle 3}})\end{array}\right]→\left[\begin{array}{rrrr}1& 0& −2& 1\\ 0& 1& \frac{{\displaystyle 4}}{{\displaystyle 3}}& −\frac{{\displaystyle 5}}{{\displaystyle 3}}\\ 0& 0& 8& −6\end{array}\right]$

Now, perform the operations $R_3=\frac{R_3}{8}$,

$\left[\begin{array}{rrrr}1& 0& −2& 1\\ 0& 1& \frac{{\displaystyle 4}}{{\displaystyle 3}}& −\frac{{\displaystyle 5}}{{\displaystyle 3}}\\ \frac{{\displaystyle 0}}{{\displaystyle 8}}& \frac{{\displaystyle 0}}{{\displaystyle 8}}& \frac{{\displaystyle 8}}{{\displaystyle 8}}& −\frac{{\displaystyle 6}}{{\displaystyle 8}}\end{array}\right]→\left[\begin{array}{rrrr}1& 0& −2& 1\\ 0& 1& \frac{{\displaystyle 4}}{{\displaystyle 3}}& −\frac{{\displaystyle 5}}{{\displaystyle 3}}\\ 0& 0& 1& −\frac{{\displaystyle 3}}{{\displaystyle 4}}\end{array}\right]$

The system of equations corresponding to this matrix is

$$\begin{gathered} \left\{x-2z=1 \\ y-\frac{4}{3}z=-\frac{5}{3} \\ z=-\frac{3}{4}\right. \end{gathered}$$

Now,

role="math" localid="1647006836563" $$\begin{gathered} x-2\left(-\frac{3}{4}\right)=1 \\ x=-\frac{1}{2} \\ y-\frac{4}{3}\left(-\frac{3}{4}\right)=-\frac{5}{3} \\ y=-\frac{5}{3}-1 \\ y=-\frac{8}{3} \end{gathered}$$

Therefore, the solutions to the given equations are$x=-\frac{1}{2},y=-\frac{8}{3},z=-\frac{3}{4}$