Chapter 1, Problem 31RE

To determine

To find: an equation of the line

Answer to Problem 31RE

The equation of the line in the general form is $x+5y=-10$ .

Explanation of Solution

Given:

  $y-\text{intercept }=-2; \text{ containing the point }\left(5,-3\right)$

Calculation:

An equation of a nonvertical line with slope m that contains the point $\left(x_1,y_1\right)$ is

  $y-y_1=m\left(x-x_1\right)$

One point on the line is given as $\left(-5,3\right)$ . Since the y -intercept is -2, another point on the line is $\left(0,-2\right)$ .

So, the slope of the line, given by $m=\frac{y-y_1}{x-x_1}$ , is $m=\frac{-3-\left(-2\right)}{5-0}=-\frac{1}{5}$ .

An equation of the line can be determined using one of the points and $m=-\frac{1}{5}$ . Choose the

point $\left(-5,3\right)$ .

  $\begin{array}{l}y-\left(-3\right)=-\frac{1}{5}\left(x-5\right)\\ y+3=-\frac{1}{5}x+1\end{array}$

Subtract 3 from both sides of the equation to isolate the constants.

  $\begin{array}{l}y+3-3=-\frac{1}{5}x+1-3\\ y=-\frac{1}{5}x-2\end{array}$

Therefore, the slope-intercept form of the line is $y=-\frac{1}{5}x-2$

In order to get the equation of the line in the general form, simplify $y=-\frac{1}{5}x-2$

Multiply by 5 on both sides of the equation.

  $\begin{array}{l}5\cdot y=5\cdot \left(-\frac{1}{5}x\right)-5\cdot 2\\ 5y=-x-10\end{array}$

Add x to both sides of the equation.

  $\begin{array}{l}x+5y=-x-10+x\\ x+5y=-10\end{array}$

Therefore, the equation of the line in the general form is $x+5y=-10$ .

Conclusion:

Therefore, the equation of the line in the general form is $x+5y=-10$ .