Chapter 3.3, Problem 52WE

To determine

To proof:Explain why $\frac{y\text{'}-y_1}{x\text{'}-x_1}=m$ .

Answer to Problem 52WE

  $m=\frac{y\text{'}-y_1}{x\text{'}-x_1}.$

Explanation of Solution

Given information:Let $Q(x\text{'},y\text{'})$ be any point of line L’ different from $P(x_1,y_1)$ .

Calculation:

Let $P(x_1,y_1)$ be a point on line L’, then the equation of the lineL’ passingthrough $P(x_1,y_1)$ having slope mis :

  $\begin{array}{c}y-y_1=m(x-x_1).\\ y-y_1=mx-m{x}_1.\\ y=mx-m{x}_1+y_1\mathrm{..................}(1)\end{array}$

Let $Q(x\text{'},y\text{'})$ be another point on line L’ , then the equation of the lineL’ passingthrough $Q(x\text{'},y\text{'})$ having slope m is:

  $\begin{array}{c}y-y\text{'}=m(x-x\text{'}).\\ y-y\text{'}=mx-mx\text{'}.\\ y=mx-mx\text{'}+y\text{'}\mathrm{..................}(2)\end{array}$

Comparing equation (1) withequation(2)

  $\begin{array}{c}mx-m{x}_1+y_1=mx-mx\text{'}+y\text{'}.\\ y\text{'}-y_1=m_{1}x\text{'}-mx.\\ y\text{'}-y_1=m(x\text{'}-x_1).\\ m=\frac{y\text{'}-y_1}{x\text{'}-x_1}.\end{array}$

Hence $m=\frac{y\text{'}-y_1}{x\text{'}-x_1}.$