System of Two Linear Equations in Matrix Form

In this tutorial we shall represent equations of straight lines into matrix form; first we will discuss one linear equation in matrix form.

One Linear Equation:
Consider the equation of straight line is given as

ax  + by + c = 0\,\,\,\,{\text{ -  -  - }}\left( {\text{i}} \right)


Equation (i) is a linear equation in two variables x and y can be written as in matrix form as follows:
Now equation (i) becomes

\begin{gathered} \Rightarrow ax + by =  - c \\ \Rightarrow \left[ {ax + by} \right] =  \left[ { - c} \right] \\ \end{gathered}


It can be further written as

\begin{gathered} \Rightarrow \left[ {\begin{array}{*{20}{c}} a&b \end{array}}  \right]\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}}  \right] = \left[ { - c} \right] \\ AX = C \\ \end{gathered}

Where A = \left[  {\begin{array}{*{20}{c}} a&b \end{array}} \right] is the coefficient matrix, X = \left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right] is variable matrix and C = \left[ { - c} \right] is the constant matrix.
Now we shall discuss system of two equations in matrix form

A System of Two Linear Equations:
Consider the system of two equations of straight lines are given as

\begin{gathered} {a_1}x + {b_1}y + {c_1} = 0\,\,\,\,{\text{  - - - }}\left( {\text{i}} \right) \\ {a_2}x + {b_2}y + {c_2} = 0\,\,\,\,{\text{  -  -   - }}\left( {{\text{ii}}} \right) \\ \end{gathered}


Equation (i) and (ii) are linear equations in two variables x and y can be written as in matrix form as follows:
Now equation (i) becomes

\begin{gathered} \Rightarrow \left\{ \begin{gathered} {a_1}x + {b_1}y =  - {c_1} \\ {a_2}x + {b_2}y =  - {c_2} \\ \end{gathered}  \right. \\ \Rightarrow \left[ {\begin{array}{*{20}{c}} {{a_1}x + {b_1}y} \\ {{a_2}x + {b_2}y} \end{array}}  \right] = \left[ {\begin{array}{*{20}{c}} { - {c_1}} \\ { - {c_2}} \end{array}}  \right] \\ \end{gathered}


It can be further written as

\begin{gathered} \Rightarrow \left[ {\begin{array}{*{20}{c}} {{a_1}}&{{b_1}} \\ {{a_2}}&{{b_2}} \end{array}}  \right]\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}}  \right] = \left[ {\begin{array}{*{20}{c}} { - {c_1}} \\ { - {c_2}} \end{array}}  \right] \\ AX = C \\ \end{gathered}


Where A = \left[  {\begin{array}{*{20}{c}} {{a_1}}&{{b_1}} \\  {{a_2}}&{{b_2}} \end{array}} \right] is the coefficient matrix, X = \left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right] is variable matrix and C = \left[ {\begin{array}{*{20}{c}} { - {c_1}} \\ { - {c_2}} \end{array}} \right] is the constant matrix.

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