Chapter 1, Problem 50SGA

To determine

To find: the standard form of line equation.

Answer to Problem 50SGA

  $2x+y+10=0$

Explanation of Solution

Given information:

  $y=-2x+\frac{1}{4}$

Coordinates = $\left(4,-8\right)$

Calculation:

A line perpendicular to line of equation, $y=-2x+\frac{1}{4}$ . So, the slope (m) is $-2$ .

The slope of line A, perpendicular to $y=-2x+\frac{1}{4}$ is opposite reciprocal of the other line. So, the slope of line can be evaluated as follows:

  $\begin{array}{c}\perp m=-\frac{1}{m}\\ -2=-\frac{1}{m}\\ m=\frac{1}{2}\end{array}$

Put the coordinates and slope value of line A, in the slope intercept equation and solve as follows:

  $\begin{array}{c}y=mx+b\\ -8=\left(\frac{1}{2}\right)\left(4\right)+b\\ -8=2+b\\ b=-8-2\\ b=-10\end{array}$

Equations for converting the slope and y-intercept into standard form as follows:

  $\begin{array}{l}m=-\frac{A}{B}\text{ and }b=-\frac{C}{B}\\ -2=-\frac{A}{B}\text{ and }-10=-\frac{C}{B}\left[\text{Put values of }m\text{ and }b\right]\\ 2=\frac{A}{B}\text{ and 10}=\frac{C}{B}\left[\text{Simplify}\right]\\ \frac{2}{1}=\frac{A}{B}\text{ and }\frac{10}{1}=\frac{C}{B}\left[\text{Finding common denominator}\right]\end{array}$

The value of $A=2,B=1,\text{ and }C=10$ .

Substitute the value of $A,B\text{ and }C$ in the standard form of line as shown below.

  $\begin{array}{l}Ax+By+C=0\\ 2x+1y+10=0\\ 2x+y+10=0\end{array}$

Thus, the standard form of line is $2x+y+10=0$ .