# System of Two Linear Equations in Matrix Form

In this tutorial we shall convert equations of straight lines into matrix form. First we will discuss one linear equation in matrix form.

__One Linear Equation__**:**

Consider the equation of a straight line is given as:

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

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

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 the system of two equations in matrix form.

__A System of Two Linear Equations__

Consider the system of two equations of straight lines is 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, and two variables, $$x$$ and $$y$$, can be written in matrix form as follows:

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

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

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