Chapter 1, Problem 124RE

To determine

To calculate:The equation for the line passing through a given point $\left(6,-3\right)$ and having slope $-\frac{1}{2}$

Answer to Problem 124RE

The equation for the line passing through $\left(6,-3\right)$ and having slope $-\frac{1}{2}$ is $2y+x=0$

Explanation of Solution

Given information:

Point $P_1=\left(-1,-6\right)$ and slope $m=-\frac{1}{2}$

Formula used:

Slope m of the line passing through two points in general say $P_1=\left(x_1,y_1\right)$ and $P_2=\left(x_2,y_2\right)$ is:

  $m=\frac{y_2-y_1}{x_2-x_1}$

Point-slope equation for a given line passing along point $P_1=\left(x_1,y_1\right)$ is:

  $y-y_1=m\left(x-x_1\right)$

Calculation:

Consider thepoint $P_1=\left(-1,-6\right)$ and slope $m=-\frac{1}{2}$

Then in order to proceed following should be known:

  $\begin{array}{l}{x}_1=6\\ y_1=-3\end{array}$

Recall the point-slope equation for a given line passing along point $P_1=\left(x_1,y_1\right)$ is:

  $y-y_1=m\left(x-x_1\right)$

Putting the values to get:

  $\begin{array}{l}y-\left(-3\right)=-\frac{1}{2}\left(x-6\right)\\ y+3=-\frac{1}{2}\left(x-6\right)\\ 2\left(y+3\right)=-1\left(x-6\right)\\ 2y+6=-x+6\\ 2y+x=0\end{array}$

Hence, the equation for the line passing through $\left(6,-3\right)$ and having slope $-\frac{1}{2}$ is $2y+x=0$