Chapter 4.2, Problem 48HP

To determine

To Calculate:

Value of p , if (3,7) , (-6,1) , (9, p ) lie on the same line.

Answer to Problem 48HP

p = 11

Explanation of Solution

Given Information:

Points (3,7) , (-6,1) , (9, p ) lie on the same line.

Concept Used:

Equation of slope-intercept form of a line passing through points $(x_1,y_1)$ and $(x_2,y_2)$ is given by:

  $y=(\frac{y_2-y_1}{x_2-x_1})x+b\mathrm{............................}(1)$

Where

b = y-intercept

  $m=slope=\frac{y_2-y_1}{x_2-x_1}$

Calculation:

First , we form the equation of line passing through points (3,7) , (-6,1):

To find slope of the line containing the given points

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\mathrm{.......................}[(x_1,y_1)=(3,7),(x_2,y_2)=(-6,1)]\\ m=\frac{1-7}{-6-3}=\frac{-6}{-9}=\frac{2}{3}\end{array}$

Use the slope and either of the points (3,7 ) , (-6,1) to find the value of p .

Let

  $\begin{array}{l}(x_1,y_1)=(3,7)\\ (x_2,y_2)=(9,p)\\ m=\frac{2}{3}=\frac{y_2-y_1}{x_2-x_1}\mathrm{......................}[from(1)]\\ \Rightarrow \frac{2}{3}=\frac{p-7}{9-3}\mathrm{..........................}[substituting\_values]\\ \Rightarrow \frac{2}{3}=\frac{p-7}{6}\mathrm{..........................}[simplify]\\ \Rightarrow \frac{2}{3}\cdot 6=\frac{p-7}{6}\cdot \mathrm{6...................}[multiply"6"each\_side]\\ \Rightarrow 4=p-\mathrm{7.............................}[simplify]\\ \Rightarrow 4+7=p-7+\mathrm{7.................}[add"7"both\_sides]\\ \Rightarrow 11=p\mathrm{................................}[simplify]\end{array}$