Chapter 4.1, Problem 5MRE

a.

To determine

The slope of the line containing the points $\left(-3,1\right),\left(5,3\right)$ .

Answer to Problem 5MRE

The value of slope is $\frac{1}{4}$ .

Explanation of Solution

Given information: The two points contained by line are $\left(-3,1\right),\left(5,3\right)$ .

Formula Used: The slope $\left(m\right)$ of line through two points $\left(x_1,y_1\right),\left(x_2,y_2\right)$ is given as,

  $m=\frac{y_2-y_1}{x_2-x_1}$

Calculation: The slope of the line through the points $\left(-3,1\right),\left(5,3\right)$ is evaluated using the formula as,

  $\begin{array}{c}m=\frac{3-1}{5+3}\mathrm{....}\left(\begin{array}{l}\left(x_1,y_1\right)\to \left(-3,1\right)\\ \left(x_2,y_2\right)\to \left(5,3\right)\end{array}\right)\\ =\frac{2}{8}\\ =\frac{1}{4}\mathrm{......}\left(1\right)\end{array}$

Hence the slope of the line is $\frac{1}{4}$ .

b.

To determine

The equation of the line in standard form through the points $\left(-3,1\right),\left(5,3\right)$ .

Answer to Problem 5MRE

The equation of line in standard form is $x-4y=-7$ .

Explanation of Solution

Given information: The two points contained by line are $\left(-3,1\right),\left(5,3\right)$ .

Formula Used: The equation of line having slope $\left(m\right)$ and through the point $\left(x_1,y_1\right)$ is given as,

  $y-y_1=m\left(x-x_1\right)$

Calculation: The equation of the line through the point $\left(-3,1\right)$ and having slope $\left(m=\frac{1}{4}\right)$ from (1) is evaluated using the formula as,

  $\begin{array}{c}y-y_1=m\left(x-x_1\right)\\ y-1=\left(\frac{1}{4}\right)\left(x+3\right)\mathrm{....}\left(\begin{array}{l}\left(x_1,y_1\right)\to \left(-3,1\right)\\ m=\frac{1}{4}\text{ from }\left(1\right)\end{array}\right)\\ 4y-4=x+\mathrm{3......}\left(\text{Cross-multiply}\right)\\ \text{or, }x-4y=-\mathrm{7......}\left(\begin{array}{l}\text{Standard form}\\ ax+by=c\end{array}\right)\end{array}$