Chapter 4.2, Problem 76S

To determine

To find: The value of $\left(r\right)$ so that the line passes through the pair of points $\left(9,r\right),\left(6,3\right)$ for which the slope $\left(m\right)$ of line is $-\frac{1}{3}$ .

Answer to Problem 76S

The value of $\left(r\right)$ is equal to $0$ .

Explanation of Solution

Given information: The given pair of points through which the line passes are $\left(9,r\right),\left(6,3\right)$ and the slope $\left(m\right)$ of line is equal to $-\frac{1}{3}$ .

Formula used: The slope $\left(m\right)$ of a line passing through the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given as,

  $\text{Slope of line}\left(m\right)=\frac{y_2-y_1}{x_2-x_1}$

Calculation: Using the formula for the slope of line, the value of $r$ can be evaluated as,

  $\begin{array}{c}\text{Slope of line}\left(m\right)=\frac{y_2-y_1}{x_2-x_1}\mathrm{......}\left(\begin{array}{l}\text{Here }m=-\frac{1}{3},\left(x_1,y_1\right)\to \left(9,r\right)\\ \text{and }\left(x_2,y_2\right)\to \left(6,3\right)\end{array}\right)\\ -\frac{1}{3}=\frac{3-r}{6-9}\mathrm{......}\left(\text{Plugging for the values}\right)\\ \left(-1\right)\left(-3\right)=3-r\mathrm{......}\left(\begin{array}{l}\text{Cross-multiplying}\\ \text{and simplifying}\end{array}\right)\\ \text{or, }3-r=3\\ r=0\end{array}$