Question

write the equation of the line that is perpendicular to the line -7 x - 9y = -35 and contains the points (-4,-7)

Answer 1

When two lines are perpendicular, Their gradients ( slopes ) are m1 x m2 = -1 , we meed to find the second gradient, then use it to find the second intercept to create the second equation of line.

Since the equation of line is y = mx + c

put the equation, -7x - 9y = -35 in the above format

-9y = -35 + 7x

$\begin{gathered} y\text{ = }\frac{-35}{-9}\text{ + }\frac{7x}{-9} \\ y\text{ = -}\frac{7x}{9}\text{ +}\frac{35}{9} \\ \text{The gradient m1 = -}\frac{7}{9} \\ m1\text{ x m2 = -1 } \\ so\text{ -}\frac{7}{9}\text{ x m2 = -1} \\ -7m2\text{ = -9} \\ m2\text{ = }\frac{9}{7} \end{gathered}$

Since the points ( -4, -7 ) are given, and gradient m2 = 9/7. We put it in the equation of line to get the intercept.

$\begin{gathered} y\text{ = mx + c } \\ -7\text{ = }\frac{9}{7}(\text{ -4 ) + c } \\ -7\text{ = }\frac{-36}{7}\text{ + c } \\ -7\text{ + }\frac{36}{7}\text{ = c} \\ \frac{-49\text{ + 36}}{7}\text{ = c } \\ \frac{-13}{7}\text{ = c} \end{gathered}$

The intercept is - 13/7 and the gradient is 9/7, hence, the equation of line is ...

$\begin{gathered} y\text{ = mx + c } \\ y\text{ = }\frac{9}{7}x\text{ - }\frac{13}{7}\text{ } \\ 7y\text{ = 9x - 13 ----This is the equation of the line. } \end{gathered}$