Chapter 3.3, Problem 54WE

To determine

To proof:Explain why L’ and L must be the same line. Therefore ,there is only one line through P having slope m.

Answer to Problem 54WE

Every point on L’ also lies on L . This can only happen when L and L’ are the same line.

Explanation of Solution

Given information:Line L through P having slope mand Line L’through Q having slope m

Calculation:

Let $P(x_1,y_1)$ be a point and m a real number .There is one and only one line L through P having slope m .Then equation of L is:

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

Let $Q(x\text{'},y\text{'})$ be a point and m a real number .There is one and only one line L’through Q having slope m .Then equation of L is:

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

From equation (1) and equation (2).

  $\begin{array}{c}mx-m{x}_1+y_1=mx-mx\text{'}+y\text{'}\\ y\text{'}-y_1.=m(x\text{'}-x_1)\mathrm{.......}(3)\end{array}$

Equation (3) shows that point $P(x_1,y_1)$ and $Q(x\text{'},y\text{'})$ both are on the same line so every point on L’ also lies on L . This can only happen when L and L’ are the same line.