Chapter 1.3, Problem 74E

a)

To determine

ToForm:The equation of the line that passes through the point $\left(-3,2\right)$ and parallel to $x+y=7$

Answer to Problem 74E

The equation of the line that passes through the point $\left(-3,2\right)$ and parallel to $x+y=7$ is $y=-x-1$

Explanation of Solution

Given:

A point $\left(-3,2\right)$ and the parallel line $x+y=7$

Concept Used:

Two distinct non vertical lines are parallel if and only of their slopes are equal.

that is $m_1=m_2$

Calculation:

Given a point $\left(-3,2\right)$ and the parallel line $x+y=7$

Finding the slope of the parallel line $x+y=7$

That is

  $x+y=7$

  $y=-x+7$

Comparing with the general slope intercept form of the equation of a line

That is $y=mx+c$ where is $m$ slope

Hence, slope of the parallel line is $m_1=-1$

Since, the lines are parallel.

So, the slope of required line is $m_2=-1$

Since, the required line passes through the point $\left(-3,2\right)$

The equation of the line with point $\left(x_1,y_1\right)$ and slope $m$ is given by

  $\left(y-y_1\right)=m\left(x-x_1\right)$

  $\begin{array}{c}\left(y-2\right)=\left(-1\right)\left(x-\left(-3\right)\right)\\ y-2=\left(-1\right)\left(x+3\right)\\ y-2=-x-3\\ y=-x-3+2\end{array}$

  $y=-x-1$

Therefore,the equation of the line that passes through the point $\left(-3,2\right)$ and parallel to $x+y=7$ is $y=-x-1$

Conclusion:

The equation of the line that passes through the point $\left(-3,2\right)$ and parallel to $x+y=7$ is $y=-x-1$

b.

To determine

ToForm:The equation of the line that passes through the point $\left(-3,2\right)$ and perpendicular to $x+y=7$

Answer to Problem 74E

The equation of the line that passes through the point $\left(-3,2\right)$ and perpendicular to $x+y=7$ is $y=x+5$

Explanation of Solution

Given:

A point $\left(-3,2\right)$ and the perpendicular line $x+y=7$

Concept Used:

Two distinct non vertical lines are perpendicular if and only of their slopes are negative reciprocals of each other.

  $m_1=-\frac{1}{m_2}$

Calculation:

Given a point $\left(-3,2\right)$ and the perpendicular line $x+y=7$

Finding the slope of the perpendicular line $x+y=7$

That is

  $x+y=7$

  $y=-x+7$

Comparing with the general slope intercept form of the equation of a line

That is

  $y=mx+c$ where is $m$ slope

Hence, slope of the perpendicular line is $m_1=-1$

Since, the lines are perpendicular.

So, the slope of required line is

  $\begin{array}{c}{m}_2=-\frac{1}{m_1}\\ =-\left(\frac{1}{-1}\right)\\ =1\end{array}$

Since, the required line passes through the point $\left(-3,2\right)$

The equation of the line with point $\left(x_1,y_1\right)$ and slope $m$ is given by

  $\begin{array}{c}\left(y-y_1\right)=m\left(x-x_1\right)\\ \left(y-2\right)=\left(1\right)\left(x-\left(-3\right)\right)\\ y-2=\left(1\right)\left(x+3\right)\\ y-2=x+3\\ y=x+3+2\\ y=x+5\end{array}$

Therefore, the equation of the line that passes through the point $\left(-3,2\right)$ and perpendicular to $x+y=7$ is $y=x+5$

Conclusion:

The equation of the line that passes through the point $\left(-3,2\right)$ and perpendicular to $x+y=7$ is $y=x+5$