# Linear Equations System Solving Task

• AKA: Solving a System of Linear Equations.
• Context:
• Example(s):
• for linear equation system $\displaystyle{ \begin{array}{lcl} 2x_1-x_2 & = & 1 \\ x_1+x_2 & = &2 \end{array} }$ Geometrically the solution is about finding a point(s) which is(are) intersection of the two given straight lines $\displaystyle{ 2x_1-x_2=1 }$ and $\displaystyle{ x_1+x_2=2 }$. The above system of linear equation in matrix form can be written as$\displaystyle{ \begin{bmatrix}2& -1\\1& 1\end{bmatrix}\begin{bmatrix} x_1\\x_2 \end{bmatrix}=\begin{bmatrix} 1\\2 \end{bmatrix} }$

• for linear equation system $\displaystyle{ \begin{array}{lcl} x_1+x_2+x_3 & = & 3 \\ x_1-x_2-x_3 & = & 1 \\ x_1+2x_2+3x_3 & = & 4 \end{array} }$ Geometrically the solution is about finding a point(s) which is(are) intersection of the three given planes $\displaystyle{ x_1+x_2+x_3=3 }$, $\displaystyle{ x_1-x_2-x_3=1 }$ and $\displaystyle{ x_1+2x_2+3x_3=4 }$. The above system of linear equation in matrix form can be written as$\displaystyle{ \begin{bmatrix}1& 1& 1\\1& -1& -1\\1& 2& 3\end{bmatrix}\begin{bmatrix} x_1\\x_2\\x_3 \end{bmatrix}=\begin{bmatrix} 3\\1\\4 \end{bmatrix} }$

• Counter-Example(s):
• See: Linear Algebra.