Chapter P, Problem 26RE

Solution

The equation of the line in general- form $Ax+By+C=0$ passing through two given points.

  $9x+7y+17=0$

Given information:

The points given are $\left(-5,4\right),\text{ and }\left(2,-5\right)$ .

Formula used:

The equation of the line passing through two points $\left(x_1,y{1}_{}\right),\text{ and }\left(x_2,y{2}_{}\right)$ is given as:

  $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

Calculation:

To find the equation of the line in point-slope form passing through point $\left(2,-1\right)$ with slope use the formula for equation passing through two points $\left(x_1,y{1}_{}\right),\text{ and }\left(x_2,y{2}_{}\right)$ that is given as:

  $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

Here, the given points are $\left(-5,4\right),\text{ and }\left(2,-5\right)$ , putting these in above formula:

  $\begin{array}{c}y-4=\frac{-5-4}{2-\left(-5\right)}\left(x-\left(-5\right)\right)\\ y=\frac{-9}{2+5}\left(x+5\right)+4\\ y=\frac{-9}{7}\left(x+5\right)+4\\ y=\frac{-9}{7}x-\frac{9}{7}\times 5+4\\ y=\frac{-9}{7}x-\frac{45}{7}+4\\ y=\frac{-9}{7}x+\frac{-45+28}{7}\\ y=\frac{-9}{7}x+\frac{-17}{7}\\ y=\frac{-9}{7}x-\frac{17}{7}\end{array}$

Now, simplify the equation $y=\frac{-9}{7}x-\frac{17}{7}$ to convert it into form $Ax+By+C=0$ as:

  $\begin{array}{c}y\times 7=\left(\frac{-9}{7}x-\frac{17}{7}\right)\times 7\\ 7y=\frac{-9}{7}\times 7x-\frac{17}{7}\times 7\\ 7y=-9x-17\\ \Rightarrow 9x+7y+17=0\end{array}$

Thus, the required equation of line is

  $9x+7y+17=0$