Step 1

Consider the following system of linear equations:

5x+8y-6z=14

3x+4y-2z=8

x+2y-2z=3

Convert above system of linear equations into augmented matrix form:

\(\displaystyle{\left[\begin{array}{ccc|c} {5}&{8}&-{6}&{14}\\{3}&{4}&-{2}&{8}\\{1}&{2}&-{2}&{3}\end{array}\right]}\)

Step 2

Transform the above matrix into reduced row echelon form:

\(\displaystyle{R}_{{2}}\rightarrow\frac{{5}}{{3}}{R}_{{2}}-{R}_{{1}}\)

\(\displaystyle{R}_{{3}}\rightarrow{5}{R}_{{3}}-{R}_{{1}}\)

\(\displaystyle{\left[\begin{array}{ccc|c} {5}&{8}&-{6}&{14}\\{0}&-\frac{{4}}{{3}}&\frac{{8}}{{3}}&-\frac{{2}}{{3}}\\{0}&{2}&-{4}&{1}\end{array}\right]}\)

\(\displaystyle{R}_{{3}}\rightarrow\frac{{2}}{{3}}{R}_{{3}}+{R}_{{2}}\)

\(\displaystyle{\left[\begin{array}{ccc|c} {5}&{8}&-{6}&{14}\\{0}&-\frac{{4}}{{3}}&\frac{{8}}{{3}}&-\frac{{2}}{{3}}\\{0}&{0}&{0}&{0}\end{array}\right]}\)

Step 3

Hence, the solution of a system of linear equations does not exist.

Consider the following system of linear equations:

5x+8y-6z=14

3x+4y-2z=8

x+2y-2z=3

Convert above system of linear equations into augmented matrix form:

\(\displaystyle{\left[\begin{array}{ccc|c} {5}&{8}&-{6}&{14}\\{3}&{4}&-{2}&{8}\\{1}&{2}&-{2}&{3}\end{array}\right]}\)

Step 2

Transform the above matrix into reduced row echelon form:

\(\displaystyle{R}_{{2}}\rightarrow\frac{{5}}{{3}}{R}_{{2}}-{R}_{{1}}\)

\(\displaystyle{R}_{{3}}\rightarrow{5}{R}_{{3}}-{R}_{{1}}\)

\(\displaystyle{\left[\begin{array}{ccc|c} {5}&{8}&-{6}&{14}\\{0}&-\frac{{4}}{{3}}&\frac{{8}}{{3}}&-\frac{{2}}{{3}}\\{0}&{2}&-{4}&{1}\end{array}\right]}\)

\(\displaystyle{R}_{{3}}\rightarrow\frac{{2}}{{3}}{R}_{{3}}+{R}_{{2}}\)

\(\displaystyle{\left[\begin{array}{ccc|c} {5}&{8}&-{6}&{14}\\{0}&-\frac{{4}}{{3}}&\frac{{8}}{{3}}&-\frac{{2}}{{3}}\\{0}&{0}&{0}&{0}\end{array}\right]}\)

Step 3

Hence, the solution of a system of linear equations does not exist.