Chapter 13.7, Problem 13CE

To determine

To state : the slope of the line and name two points on the line

Answer to Problem 13CE

The slope of the line is $\frac{a}{b}$ and the two points are $\left(0,-\frac{ad+bc}{b}\right)$ , $\left(1,\frac{a\left(1-d\right)+bc}{b}\right)$

Explanation of Solution

Given information :

  $y-c=\frac{a}{b}\left(x-d\right)$

Formula used :

  $y=mx+b$

Calculation :

The value of $y$

  $\begin{array}{l}y-c=\frac{a}{b}\left(x-d\right)\\ y=\frac{a}{b}x-\frac{ad}{b}+c\\ y=\frac{a}{b}x-\frac{ad+bc}{b}\end{array}$

Let the slope of the line $=m$

The equation is $y=\frac{a}{b}x-\frac{ad}{b}+c$

Therefore, the slope of the line $=\frac{a}{b}=m$

Let the point $x=0$

Therefore, the equation is $y=-\frac{ad+bc}{b}$

Let the point $x=1$

Therefore, the equation is

  $\begin{array}{l}y=\frac{a}{b}\cdot 1-\frac{ad+bc}{b}\\ y=\frac{a}{b}-\frac{ad+bc}{b}\\ y=\frac{a-ad+bc}{b}\\ y=\frac{a\left(1-d\right)+bc}{b}\end{array}$

Therefore, the first and second point are $\left(0,-\frac{ad+bc}{b}\right)$ and $\left(1,\frac{a\left(1-d\right)+bc}{b}\right)$

Hence, the slope of the line is $\frac{a}{b}$ and the two points are $\left(0,-\frac{ad+bc}{b}\right)$ , $\left(1,\frac{a\left(1-d\right)+bc}{b}\right)$