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Chapter 1: Problem 74

Write equations of the lines through the given point (a) parallel to and (b) perpendicular to the given line. $$x+y=7, \quad(-3,2)$$

Short Answer

Expert verified
The equation of line parallel to the given line through the given point is \(y=-x+5\). The equation of line perpendicular to the given line through the given point is \(y=x+5\).

Step by step solution

01

Find the slope of the given line

First, rearrange the given line equation into slop-intercept form \(y=mx+c\). This will give us \(x+y=7 \) to \( y=-x+7\). Thus, from this equation we can identify the slope \(m\) of the given line as -1.
02

Find the parallel line

The line parallel to the given line will have the same slope as the given line. Hence, the slope is -1. Substituting into the line equation \(y-y_1=m(x-x_1)\), where \((x_1, y_1)\) is the given point \((-3,2)\), then we get \(y-2=-1(x+3)\). Solving for \(y\) gives the equation \(y=-x+5\).
03

Find the perpendicular line

The perpendicular line to the given line will have the slope which is the negative reciprocal of -1, which is \(m = 1\). Substituting into the line equation \(y-y_1=m(x-x_1)\) as done previously then gives us, \(y-2=1(x+3)\). Solving for \(y\) gives the equation \(y=x+5\).

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