Chapter B, Problem 21E

To determine

To find: The equation of the line for the given conditions.

Answer to Problem 21E

The equation of the line is $y=\frac{3}{4}x-6$.

Explanation of Solution

Given:

The required line is passing through the points $\left(1,-6\right)$ and parallel to $x+2y=6$.

Property used:

Two non-vertical lines are parallel if and only if they have the same slope.

The equation of the line with slope $m$ and $y\text{ intercept }b\text{ is}$, $y=mx+b$.

The equation of the line passes through the point $\left(x_1,y_1\right)$ and having slope m is, $y-y_1=m\left(x-x_1\right)$.

Calculation:

To find the slope, rewrite the equation,

$\begin{array}{l}x+2y=6\\ 2y=6-x\\ y=\frac{6}{2}-\frac{1}{2}x\\ y=-\frac{1}{2}x+3\\ y=mx+b\end{array}$

Therefore, $m=-\frac{1}{2}$.

Substitute the values in the above mentioned property,

$\begin{array}{l}y-y_1=m\left(x-x_1\right)\\ y-\left(-6\right)=-\frac{1}{2}\left(x-1\right)\\ y+6=-\frac{1}{2}\left(x-1\right)\\ y+6=-\frac{1}{2}x+\frac{1}{2}\\ y=-\frac{1}{2}x+\frac{1}{2}-6\\ y=-\frac{1}{2}x+\left(\frac{1}{2}-\frac{6\times 2}{2}\right)\\ y=-\frac{1}{2}x+\left(\frac{1}{2}-\frac{12}{2}\right)\\ y=-\frac{1}{2}x-\frac{11}{2}\end{array}$

Thus, the equation of the line is $y=-\frac{1}{2}x-\frac{11}{2}$.