Chapter 4.3, Problem 10CYU

To determine

The equation of line GH in point slope form for the graph shown below.

  

Answer to Problem 10CYU

The equation is $y-7=-\frac{6}{7}\left(x+3\right)$ .

Explanation of Solution

Given:

From the given graph the vertices of line GH are $G\left(-3,7\right)\text{ and }H\left(4,1\right)$

Calculation:

The slope $\left(m_{GH}\right)$ for the line GH through the points $G\left(-3,7\right)\text{ and }H\left(4,1\right)$ using slope formula is evaluated as,

  $\begin{array}{c}\left(m_{GH}\right)=\frac{1-7}{4+3}\\ =-\frac{6}{7}\end{array}$

Now, the equation for line GH having slope $\left(m_{GH}=-\frac{6}{7}\right)$ and passing through the point $G\left(-3,7\right)$ is evaluated as,

  $\begin{array}{c}y-y_1=m_{GH}\left(x-x_1\right)\mathrm{......}\left(\text{Point-slope form}\right)\\ y-7=-\frac{6}{7}\left(x+3\right)\mathrm{......}\left(\begin{array}{l}\text{Here }{m}_{GH}=-\frac{6}{7}\\ \left(x_1,y_1\right)\to \left(-3,7\right)\end{array}\right)\\ y-7=-\frac{6}{7}\left(x+3\right)\end{array}$

To determine

The equation of line GH in standard form for the graph shown below.

  

Answer to Problem 10CYU

The equation in standard form is $6x+7y=31$ .

Explanation of Solution

Given:

From the given graph the vertices of line GH are $G\left(-3,7\right)\text{ and }H\left(4,1\right)$

Calculation:

The slope $\left(m_{GH}\right)$ for the line GH through the points $G\left(-3,7\right)\text{ and }H\left(4,1\right)$ using slope formula is evaluated as,

  $\begin{array}{c}\left(m_{GH}\right)=\frac{1-7}{4+3}\\ =-\frac{6}{7}\end{array}$

Now, the equation for line GH having slope $\left(m_{GH}=-\frac{6}{7}\right)$ and passing through the point $G\left(-3,7\right)$ is evaluated as,

  $\begin{array}{c}y-y_1=m_{GH}\left(x-x_1\right)\mathrm{......}\left(\text{Point-slope form}\right)\\ y-7=-\frac{6}{7}\left(x+3\right)\mathrm{......}\left(\begin{array}{l}\text{Here }{m}_{GH}=-\frac{6}{7}\\ \left(x_1,y_1\right)\to \left(-3,7\right)\end{array}\right)\\ 7\left(y-7\right)=-6\left(x+3\right)\\ 7y-49=-6x-18\\ \text{or, }6x+7y=\mathrm{31......}\left(\text{Standard form}\right)\end{array}$