The equations, 2x – 7y + 11 = 0 and x + 3y – 8 = 0
The equation of the straight line passing through the points of intersection of 2x − 7y + 11 = 0 and x + 3y − 8 = 0 is given below:
![\[\begin{array}{*{35}{l}} 2x~-~7y\text{ }+\text{ }11\text{ }+~\lambda \left( x\text{ }+\text{ }3y~-~8 \right)\text{ }=\text{ }0 \\ \left( 2\text{ }+~\lambda \right)x\text{ }+\text{ }\left( -~7\text{ }+\text{ }3\lambda \right)y\text{ }+\text{ }11~-~8\lambda ~=\text{ }0 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4b076bdbdb640b4a104265f6200880da_l3.png "Rendered by QuickLaTeX.com")
(i) The required line is parallel to the x-axis. So, the coefficient of x should be zero.
![\[\begin{array}{*{35}{l}} 2\text{ }+~\lambda ~=\text{ }0 \\ \lambda ~=\text{ }-2 \\ {} \\ 0\text{ }+\text{ }\left( -~7~-~6 \right)y\text{ }+\text{ }11\text{ }+\text{ }16\text{ }=\text{ }0 \\ 13y~-~27\text{ }=\text{ }0 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-33a37bbd45346a6fd0c5202b7c5b8015_l3.png "Rendered by QuickLaTeX.com")
∴ The equation of the required line is 13y − 27 = 0
(ii) The required line is parallel to the y-axis. So, the coefficient of y should be zero.
![\[<span class="ql-right-eqno"> (1) </span><span class="ql-left-eqno"> </span><img src="https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-17b883c8fddbd3a25dde7ee0c81a1076_l3.png" height="114" width="581" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{align*} & -7\text{ }+\text{ }3\lambda ~=\text{ }0\lambda \text{ }=\text{ }7/3 \\ & \begin{array}{*{35}{l}} \left( 2\text{ }+\text{ }7/3 \right)x\text{ }+\text{ }0\text{ }+\text{ }11\text{ }-\text{ }8\left( 7/3 \right)\text{ }=\text{ }0 \\ 13x\text{ }\text{ }23\text{ }=\text{ }0 \\ \end{array} \\ \end{align*}" title="Rendered by QuickLaTeX.com"/>\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-685aa6082064ae59127e5dda4d7fafbd_l3.png "Rendered by QuickLaTeX.com")
∴ The equation of the required line is 13x – 23 = 0