Use the​ Gauss-Jordan method to solve the system of equations. If the system has infinitely many​ solutions, give the solution with z arbitrary. x-5y+2z=1 3x-4y+2z=-1

Question
Equations
asked 2021-02-19
Use the​ Gauss-Jordan method to solve the system of equations. If the system has infinitely many​ solutions, give the solution with z arbitrary.
x-5y+2z=1
3x-4y+2z=-1

Answers (1)

2021-02-20
Step 1
To find the solution of system of equations reduce the matrix \(\displaystyle{\left[{A}{\mid}{B}\right]}\) where A is the matrix formed by the coefficients of LHS of the equations and B is the matrix formed by the RHS of the equations.
Step 2
Find the matrix \(\displaystyle{\left[{A}{\mid}{B}\right]}\) from the given system of equations and row reduce it.
\(\displaystyle{\left[\begin{array}{ccc|c} {1}&-{5}&{2}&{1}\\{3}&-{4}&{2}&-{1}\end{array}\right]}{R}_{{2}}\rightarrow{R}_{{2}}-{3}{R}_{{1}}\)
\(\displaystyle{\left[\begin{array}{ccc|c} {1}&-{5}&{2}&{1}\\{0}&{11}&-{4}&-{4}\end{array}\right]}{R}_{{2}}\rightarrow\frac{{R}_{{2}}}{{11}}\)
\(\displaystyle{\left[\begin{array}{ccc|c} {1}&-{5}&{2}&{1}\\{0}&{1}&-\frac{{4}}{{11}}&-\frac{{4}}{{11}}\end{array}\right]}{R}_{{1}}\rightarrow{R}_{{1}}+{5}{R}_{{2}}\)
\(\displaystyle{\left[\begin{array}{ccc|c} {1}&{0}&\frac{{2}}{{11}}&-\frac{{9}}{{11}}\\{0}&{1}&-\frac{{4}}{{11}}&-\frac{{4}}{{11}}\end{array}\right]}\)
Step 3
Form the equation from the above matrix.
\(\displaystyle{x}+\frac{{2}}{{11}}{z}=-\frac{{9}}{{11}}\)
\(\displaystyle{y}-\frac{{4}}{{11}}{z}=-\frac{{4}}{{11}}\)
Step 4
Let z=t. Find the value of x and y.
\(\displaystyle{x}+\frac{{2}}{{11}}{\left({t}\right)}=-\frac{{9}}{{11}}\)
\(\displaystyle{x}=-\frac{{9}}{{11}}-\frac{{{2}{t}}}{{11}}\)
\(\displaystyle{y}-\frac{{4}}{{11}}{\left({t}\right)}=-\frac{{4}}{{11}}\)
\(\displaystyle{y}=-\frac{{4}}{{11}}+\frac{{{4}{t}}}{{11}}\)
Step 5
Answer: For random z=t the solution set will be,
\(\displaystyle{\left\lbrace{\left(\frac{{-{9}}}{{11}}-\frac{{{2}{t}}}{{11}},\frac{{-{4}}}{{11}}+\frac{{{4}{t}}}{{11}}\right)},{t}\right.}\rbrace\)
0

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