 # Use Gaussian elimination to find the complete solution to the system of given equations, or show that none exists. {(2x-4y+z=3),(x-3y+z=5),(3x-7y+2z=12):} Alyce Wilkinson 2021-01-19 Answered
Use Gaussian elimination to find the complete solution to the system of given equations, or show that none exists.
$\left\{\begin{array}{c}2x-4y+z=3\\ x-3y+z=5\\ 3x-7y+2z=12\end{array}$
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it ottcomn
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.