# Neutrosophic Linear Equations and Application in Traffic Flow Problems
## 4 Solving a System of Neutrosophic Linear Equations  
To determine if such a system of neutrosophic linear equations has a solution, one needs to find out if there exists a group of numbers for \(x_{1}\), \(x_{2}\),..., \(x_{n}\) which can satisfy each of the equations in Equation (2) simultaneously. The system is consistent if it has a solution. Otherwise, the system is inconsistent. Generally, a system of neutrosophic linear equations can be solved by solving methods of systems of classical linear equations, such as the substitution method, the addition method, and the inversion matrix method. In this section, two examples of the system of neutrosophic linear equations show solving processes by using the substitution method, the addition method, and the inverse of a square matrix. **Example 2**. : _Let us consider the following system of neutrosophic linear equations in two variables:_  
\[3 lx_{1}+2x_{2}=2\ (\text{the first equation}),\]  
\[5 lx_{1}+7x_{2}=3\ (\text{the second equation}).\]  
Then, we use the substitution method, the addition method, and the inverse matrix method, respectively, to solve the system of neutrosophic linear equations. 1. Using the substitution method Solve the first equation for \(x_{1}\): \[x_{1}=(2-2 lx_{2})/(3l).\] Substitute \(x_{1}=(2-2 lx_{2})/(3l)\) into the second equation: \[5l(2-2 lx_{2})/(3l)+7x_{2}=3.\] Then, \(5(2-2 lx_{2})/3+7x_{2}=3.\) Thus, the solution for \(x_{2}\) is \(x_{2}=-1/11\), and then we obtain \(3 lx_{1}-2/11=2\) from the first equation. Hence, the solution of the system is \(x_{1}=8/(11l)\) and \(x_{2}=-1/11\). 2. Using the addition method Multiply the first equation by \(-5\) and the second equation by \(3\): \[-15 lx_{1}-10x_{2}=-10,\] \[15 lx_{1}+21x_{2}=9.\] Add the two equations: \[11x_{2}=-1.\] Thus, \(x_{2}=-1/11\), and then \(3 lx_{1}-2/11=2\) for the first equation. Therefore, the solution of the system is \(x_{1}=8/(11l)\) and \(x_{2}=-1/11\). 3. Using the inverse matrix method If \(AX=B\) is a neutrosophic matrix equation composed of \(m\) neutrosophic linear equations in \(n\) unknowns and if the inverse \(A^{-1}\) exists (i.e., \(A\) is nonsingular), by multiplying both sides of the equation \(AX=B\) by \(A^{-1}\) (on the left), we obtain the desired solution to the neutrosophic matrix equation as \(X=A^{-1}B\). Let us write the system in Example 2 by the neutrosophic matrices:  
\[A=\left[ \begin{array}{cc}3I&2\\ 5I&7\end{array} \right],\ X=\left[ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right],\ \text{and}\ B=\left[ \begin{array}{c}2\\ 3\end{array} \right].\]  
Then, the system of neutrosophic linear equations can be written in the form of the neutrosophic matrix equation:  
\[AX=B. \tag{4}\]  
Since the determinant of the neutrosophic matrix \(A\) can be evaluated by \(\left|A\right|=21I-10I=11I\), the inverse of the neutrosophic matrix \(A\) is as follows:  
\[A^{-1}=\left[ \begin{array}{cc}7/(11I)&-2/(11I)\\ -5/11&3/11\end{array} \right].\]Using this result, by multiplying both sides of Equation (4) by \(A^{-1}\) (on the left), we find that the solution of the system is  
\[X=A^{-1}B=\left[ \begin{array}{cc}7/(11I)&-2/(11I)\\ -5/11&3/11\end{array} \right]\left[ \begin{array}{c}2\\ 3\end{array} \right]=\left[ \begin{array}{c}8/(11I)\\ -1/(11)\end{array} \right].\]  
Hence, the solution of the system is \(x_{1}=8/(11I)\) and \(x_{2}=-1/11\). Especially, if \(I\in[2,3]\) is considered as a possible interval of the indeterminacy \(I\), then the solution of the system is \(x_{1}=[0.2424,0.3636]\) and \(x_{2}=-0.0909\). If \(I=2\), the system is degenerated to the system of classical linear equations, and its solution is given by \(x_{1}=0.3636\) and \(x_{2}=-0.0909\). It is obvious that the system of neutrosophic linear equations may be written in a compact form with the help of matrices. Hence, one will use this matrix equation representation and apply the inverse matrix method for solving this neutrosophic matrix equation easily by the MATLAB software.