Equation of Line Parallel to another Line

Consider we have given equation of a line and this line is parallel to another line which is passes through any given point and to find the required equation of line with the help of given line.
Now we explain this concept with the help of an example

Example: Find the equation of straight line passing through the point \left( {1,2} \right) and is parallel to another given line whose equation is 2x - 3y  + 5 = 0.


First find the slope of given line by comparing slope intercept form of a line as follows:

\begin{gathered} 2x - 3y + 5 = 0 \\ \Rightarrow 3y = 2x + 5 \\ \Rightarrow y = \frac{2}{3}x + \frac{5}{3} \\ \end{gathered}

Compare with slope intercept form, to find the slope of line y = mx + c
Now slope of given line m  = \frac{2}{3}, since given line is parallel to the required line and we know that slope of parallel are the same, i.e. under the condition of parallel lines {m_1} = {m_2}
Now slope of required is also equals to m = \frac{2}{3}, since the required line passes through the given point \left( {1,2}  \right). To find equation of required line using slope point form is given by

\begin{gathered} y - {y_1} = m\left( {x - {x_1}} \right) \\ \Rightarrow y - 2 = \frac{2}{3}\left( {x -  1} \right) \\ \Rightarrow 3y - 6 = 2x - 2 \\ \Rightarrow 2x - 3y + 4 = 0 \\ \end{gathered}

Which is the equation of straight line parallel to the line 2x - 3y + 5 = 0. In this calculated equation observe that the coefficients of x and y are same.