# System of Linear Equations

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• AKA: SLE.
• Context:
• Example(s):
• $\displaystyle{ \begin{array}{lcl} 2x_1-x_2 & = & 1 \\ x_1+x_2 & = &2 \end{array} }$. This is a consistent system with unique solution $\displaystyle{ x_1 = 1, x_2 = 1. }$

• $\displaystyle{ \begin{array}{lcl} x_1+x_2 & = & 2 \\ 2x_1+2x_2 & = &4 \end{array} }$. This is a consistent system with infinitely many solutions. One of the solution of the system is $\displaystyle{ x_1 = 1, x_2 = 1. }$

• $\displaystyle{ \begin{array}{lcl} x_1+x_2 & = & 2 \\ x_1+x_2 & = &1 \end{array} }$. This is an inconsistent system. So no solution exist for the system.

• $\displaystyle{ \begin{array}{lcl} x_1+x_2+x_3 & = & 3 \\ x_1-x_2-x_3 & = & 1 \\ x_1+2x_2+3x_3 & = & 4 \end{array} }$. This system is consistent with a unique solution $\displaystyle{ x_1 = 2, x_2 = 1, x_3= 0 . }$

• \displaystyle{ \begin{alignat}{7} 3x &&\; + \;&& 2y &&\; - \;&& z &&\; = \;&& 1 & \\ 2x &&\; - \;&& 2y &&\; + \;&& 4z &&\; = \;&& -2 & \\ -x &&\; + \;&& \tfrac{1}{2} y &&\; - \;&& z &&\; = \;&& 0 & \end{alignat}. }This system is consistent with a unique solution $\displaystyle{ x_1 = 1, x_2 = -2, x_3= -2 . }$
• Counter-Example(s):
• See: Linear Function, Formal System, Mathematical Model, Linear Algebra, Equation Solving, Numerical Linear Algebra.