Chapter 12, Problem 76CLR

To determine

To write: An equation in standard form of the line containing $\left(-1,1\right)$ and $\left(6,-1\right)$ .

Answer to Problem 76CLR

The equation of the line is $2x+7y=5$ .

Explanation of Solution

Given:

The points are $\left(-1,1\right)$ and $\left(6,-1\right)$ .

Calculation:

For any two given pints $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ , the equation of the line containing the two points is given by $y=mx+c$ , where c is the y -intercept of the line and m is the slope of the line given by $m=\frac{y_2-y_1}{x_2-x_1}$ .

So, the slope of the line passing through $\left(-1,1\right)$ and $\left(6,-1\right)$ is,

  $\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{-1-1}{6-\left(-1\right)}\\ =\frac{-2}{6+1}\\ =-\frac{2}{7}\end{array}$

In order to determine the value of c , substitute $-\frac{2}{7}$ for m , -1 for x and 1 for y in the equation $y=mx+c$ and simplify.

  $\begin{array}{c}y=mx+c\\ 1=\left(-\frac{2}{7}\right)\left(-1\right)+c\\ 1=\frac{2}{7}+c\\ \frac{5}{7}=c\left[\text{Subtracting }\frac{2}{7}\text{ from both sides}\right]\end{array}$

So, the equation of the line becomes $y=-\frac{2}{7}x+\frac{5}{7}$ .

Since the standard form of a line is of the form $ax+by=c$ , where a , b and c are integers, multiply both sides of the equation $y=-\frac{2}{7}x+\frac{5}{7}$ by 7 and simplify.

  $\begin{array}{c}7y=-2x+5\\ 2x+7y=2x-2x+5\left[\text{Adding }2x\text{ on both sides}\right]\\ 2x+7y=5\end{array}$

So, the standard form of the equation is $2x+7y=5$ .