Chapter 7, Problem 9CT

To determine

To solve: the system using matrices.

Answer to Problem 9CT

Therefore, the solution of given system of equations is $x=\frac{4-4k-1}{2},y=1+2k,z=k$ .

Explanation of Solution

Given information :

System of equations is given as,

  $\begin{array}{l}2x+y+2z=4\\ 2x+2y=5\\ 2x-y+6z=2\end{array}$

Writing the given system of equations in the form Ax = B.

  $\left[\begin{array}{ccc}2& 1& 2\\ 2& 2& 0\\ 2& -1& 6\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}4\\ 5\\ 2\end{array}\right]$

So, converting above matrix in echelon form,

  $\left[\begin{array}{ccc}2& 1& 2\\ 2& 2& 0\\ 2& -1& 6\end{array}\begin{array}{c}:\\ :\\ :\end{array}\begin{array}{c}4\\ 5\\ 2\end{array}\right]$

Further, applying row transformations,

  $\begin{array}{l}\left[\begin{array}{ccc}2& 1& 2\\ 2& 2& 0\\ 2& -1& 6\end{array}\begin{array}{c}:\\ :\\ :\end{array}\begin{array}{c}4\\ 5\\ 2\end{array}\right]\stackrel{R_2\to R_2-R_1}{\to }\left[\begin{array}{ccc}2& 1& 2\\ 0& 1& -2\\ 2& -1& 6\end{array}\begin{array}{c}:\\ :\\ :\end{array}\begin{array}{c}4\\ 1\\ 2\end{array}\right]\\ \left[\begin{array}{ccc}2& 1& 2\\ 0& 1& -2\\ 2& -1& 6\end{array}\begin{array}{c}:\\ :\\ :\end{array}\begin{array}{c}4\\ 1\\ 2\end{array}\right]\stackrel{R_3\to R_3-R_1}{\to }\left[\begin{array}{ccc}2& 1& 2\\ 0& 1& -2\\ 0& -2& 4\end{array}\begin{array}{c}:\\ :\\ :\end{array}\begin{array}{c}4\\ 1\\ -2\end{array}\right]\\ \left[\begin{array}{ccc}2& 1& 2\\ 0& 1& -2\\ 0& -2& 4\end{array}\begin{array}{c}:\\ :\\ :\end{array}\begin{array}{c}4\\ 1\\ -2\end{array}\right]\stackrel{R_3\to R_3+2R_1}{\to }\left[\begin{array}{ccc}2& 1& 2\\ 0& 1& -2\\ 0& 0& 0\end{array}\begin{array}{c}:\\ :\\ :\end{array}\begin{array}{c}4\\ 1\\ 0\end{array}\right]\end{array}$

From the last matrix, it is clear that the matrix has infinitely many solutions.

So,

Let $z=k$ ,

Then $y=1+2k$

And,

  $\begin{array}{c}2x+y+2z=4\\ 2x+\left(1+2k\right)+2k=4\\ 2x+1+4k=4\\ 2x=4-4k-1\\ x=\frac{4-4k-1}{2}\end{array}$

Therefore, the solution of given system of equations is $x=\frac{4-4k-1}{2},y=1+2k,z=k$ .