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Chapter 2: Problem 17

a. Write an equation for the line parallel to \(y=4-x\) that passes through the point (3,7) . b. Find an equation for the line perpendicular to \(y=4-x\) that passes through the point (3,7) .

Short Answer

Expert verified
a. y = -x + 10; b. y = x + 4

Step by step solution

01

Understand the Slope

Determine the slope of the given line. The equation of the line is given in the form: y = 4 - x. This can be rewritten as y = -x + 4. Therefore, the slope (m) of the line is -1.
02

Step 1a: Write the Equation of the Parallel Line

A parallel line will have the same slope as the original line. Using the point-slope form of the equation: y - y_1 = m(x - x_1), where (x_1, y_1) is the point (3, 7) and m = -1. Thus, y - 7 = -1(x - 3).
03

Step 2a: Simplify the Equation

Simplify the equation to get it into the slope-intercept form y = mx + b: y - 7 = -x + 3, then y = -x + 10.
04

Find the Slope of the Perpendicular Line

The slope of the line perpendicular to the original line is the negative reciprocal of -1. Therefore, the slope is 1.
05

Step 1b: Write the Equation of the Perpendicular Line

Again, using the point-slope form of the equation: y - y_1 = m(x - x_1), where (x_1, y_1) is the point (3, 7) and m = 1. Thus, y - 7 = 1(x - 3).
06

Step 2b: Simplify the Equation

Simplify the equation to get it into the slope-intercept form y = mx + b: y - 7 = x - 3, then y = x + 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parallel Lines
Parallel lines are lines in a plane that never meet. They always have the same slope. For example, consider the line given by the equation \( y = 4 - x \). This line can be rewritten in slope-intercept form as \( y = -x + 4 \), showing a slope of -1. Any line parallel to this will also have a slope of -1. So, to find the equation of a parallel line passing through a specific point, say (3,7), we use the point-slope form equation: \( y - y_1 = m(x - x_1) \). Substituting in the point (3,7) and the slope -1, we get: \( y - 7 = -1(x - 3) \). Simplifying this, we achieve: \( y = -x + 10 \). So, the parallel line passes through (3,7) and has the same slope as the original line.
Perpendicular Lines
Perpendicular lines intersect at a right angle (90 degrees). The slopes of two perpendicular lines are negative reciprocals of each other. For the given line \( y = 4 - x \), the slope is -1. The negative reciprocal of -1 is 1. To find the equation of a line perpendicular to the original line and passing through the point (3,7), we use the point-slope equation again: \( y - y_1 = m(x - x_1) \). Substituting the point (3,7) and the new slope 1, we have: \( y - 7 = 1(x - 3) \). Simplifying this expression leads to: \( y = x + 4 \). Thus, this line is perpendicular to the original line and runs through the point (3,7).
Point-Slope Form
The point-slope form of a linear equation is a practical tool for finding the equation of a line when you know its slope and one point on the line. The form is expressed as: \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line, and \( m \) is the line's slope. This form is directly used in problems involving parallel and perpendicular lines. For example, to write the equation of a line with a slope of -1 passing through the point (3,7), you'd plug in the values to get: \( y - 7 = -1(x - 3) \). Simplifying it would then yield the equation in slope-intercept form.
Slope-Intercept Form
The slope-intercept form of a linear equation is the most common way to express a line equation: \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept. It's practical for quickly identifying the slope and y-intercept of a line. For instance, the line \( y = 4 - x \) can be rewritten as \( y = -x + 4 \), revealing a slope (\( m \)) of -1 and a y-intercept (\( b \)) of 4. Converting from point-slope to slope-intercept form can help in understanding the line's behavior. Simplifying \( y - 7 = -x + 3 \) leads to \( y = -x + 10 \), clearly identifying the slope and y-intercept of the desired line.

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