Chapter 9.2, Problem 31HP

To determine

To find: The equation of the translated line.

Answer to Problem 31HP

The equation of the translated line is $y=mx-am+2b$ .

Explanation of Solution

Given information:

The given information is a line $y=mx+b$ is translated using the vector $\left(a,b\right)$ .

Calculation:

We are given the line:

  $y=mx+b$

The translation moves each point $\left(x,y\right)$ of the line in the point $\left(x+a,y+b\right)$ :

  $\begin{array}{l}y=0\\ 0=mx+b\\ x=-\frac{b}{m}\end{array}$

  $\begin{array}{l}x=0\\ y=m\times 0+b\\ y=b\\ x=-\frac{b}{m}\end{array}$

  $\begin{array}{l}A\left(0,b\right)\\ B\left(-\frac{b}{m},0\right)\end{array}$

We determine the x and y intercepts of the original line:

  $\begin{array}{l}A\left(0,b\right)\to A\text{'}\left(0+a,b+b\right)=\left(a,2b\right)\\ B\left(-\frac{b}{m},0\right)\to B\text{'}\left(-\frac{b}{m}+a,0+b\right)=\left(a-\frac{b}{m},b\right)\end{array}$

We use the points $A$ , and $B\text{'}$ to determine the equation of the line's translation:

  $m\text{'}=\frac{2b-b}{a-\left(a-\frac{b}{m}\right)}=\frac{b}{\frac{b}{m}}=m$

  $\begin{array}{l}y-yA\text{'}=m\text{'}\left(x-xA\text{'}\right)\\ y-2b=\left(x-a\right)\\ y=mx-am+2b\end{array}$

The $x$ and $y$ intercepts of the line’s translation are:

  $\begin{array}{c}x=0\\ y=m\times 0-am+2b\\ =2b-am\end{array}$

  $\begin{array}{c}y=0\\ 0=m\times x-am+2b\\ x=a-\frac{2b}{m}\end{array}$

Therefore, the equation of the translated line is $y=mx-am+2b$ .