Alyce Wilkinson

2021-01-19

Use Gaussian elimination to find the complete solution to the system of given equations, or show that none exists.
$\left\{\begin{array}{c}2x-4y+z=3\\ x-3y+z=5\\ 3x-7y+2z=12\end{array}$

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Step 1
Consider the system of equations,
$\left\{\begin{array}{c}2x-4y+z=3\\ x-3y+z=5\\ 3x-7y+2z=12\end{array}$
Step 2
The augmented matrix is constructed as below.
$\left[\begin{array}{cccc}2& -4& 1& 3\\ 1& -3& 1& 5\\ 3& -7& 2& 12\end{array}\right]$
Use Gaussian elimination to find the complete solution to the system of given equations.
Apply the row operation ${R}_{1}↔{R}_{2}$
$\left[\begin{array}{cccc}1& -3& 1& 5\\ 2& -4& 1& 3\\ 3& -7& 2& 12\end{array}\right]$
Apply the row operation ${R}_{2}\to {R}_{2}-2{R}_{1}$ to the above matrix.
$\left[\begin{array}{cccc}1& -3& 1& 3\\ 0& 2& -1& -7\\ 3& -7& 2& 12\end{array}\right]$
Aply the row operation ${R}_{3}\to {R}_{3}-3{R}_{1}$ to the above matrix.
$\left[\begin{array}{cccc}1& -3& 1& 3\\ 0& 2& -1& -7\\ 0& 2& -1& -3\end{array}\right]$
Aply the row operation ${R}_{2}\to \frac{1}{2}{R}_{2}$ to the above matrix.
$\left[\begin{array}{cccc}1& -3& 1& 5\\ 0& 1& -\frac{1}{2}& -\frac{7}{2}\\ 0& 2& -1& -3\end{array}\right]$
Apply the row operation ${R}_{2}\to {R}_{3}-2{R}_{2}$ to the above matrix.
$\left[\begin{array}{cccc}1& -3& 1& 5\\ 0& 1& -\frac{1}{2}& -\frac{7}{2}\\ 0& 0& 0& 4\end{array}\right]$
Translate the last row back in to equation form as, 0x+0y+0z=4 which is false equation since it is impossible to find the solution to the given system of equations.
Thus, the system has no solution.
Therefore, the solution set of the system of equations is an empty set.

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