/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 Given below are descriptions of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 39

Given below are descriptions of two lines. Find the slopes of Line 1 and Line \(2 .\) Is each pair of lines parallel, perpendicular or neither? Line 1: Passes through (0,5) and (3,3) Line 2: Passes through (1,-5) and (3,-2)

Short Answer

Expert verified
The lines are perpendicular.

Step by step solution

01

Find the Slope of Line 1

The slope of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For Line 1, the given points are (0, 5) and (3, 3). Use the formula to find the slope: \[ m_1 = \frac{3 - 5}{3 - 0} = \frac{-2}{3} = -\frac{2}{3} \]. Thus, the slope of Line 1 is \(-\frac{2}{3}\).
02

Find the Slope of Line 2

Using the same formula, the slope of Line 2 is determined by the points \( (1, -5) \) and \( (3, -2) \). Plug the points into the slope formula: \[ m_2 = \frac{-2 - (-5)}{3 - 1} = \frac{-2 + 5}{2} = \frac{3}{2} \]. So, the slope of Line 2 is \(\frac{3}{2}\).
03

Determine if the Lines are Parallel, Perpendicular, or Neither

Two lines are parallel if they have the same slope, and they are perpendicular if the product of their slopes is \(-1\). Compare the slopes of Line 1 and Line 2. For Line 1, \( m_1 = -\frac{2}{3} \) and for Line 2, \( m_2 = \frac{3}{2} \). Compute the product of these slopes: \[ m_1 \times m_2 = -\frac{2}{3} \times \frac{3}{2} = -1 \]. Since the product is \(-1\), the lines are perpendicular.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Parallel Lines
Parallel lines are a fundamental concept in geometry and algebra. They are lines in a plane that never intersect. This means they will continue endlessly in both directions without ever meeting. A key property of parallel lines is that they have the same slope.

For example, if one line has a slope of \(m = 4\), another line parallel to it will also have a slope of \(m = 4\). This consistency in slope signifies that their steepness, or gradient, is identical.

However, it is important to note that parallel lines do not need to have the same y-intercept. They can be shifted vertically anywhere along the plane. Remember, if two lines have different slopes, they cannot be parallel.
  • Parallel lines: Same slope.
  • Never intersect.
  • Different or same y-intercepts.
Perpendicular Lines
Perpendicular lines intersect at right angles (90 degrees). When you think of perpendicular lines, think of a perfect cross.

A unique property of perpendicular lines is their slopes. The slopes of two perpendicular lines are negative reciprocals of each other. Negative reciprocal means that if one line's slope is \(a\), the slope of the perpendicular line will be \(-\frac{1}{a}\).

In the problem, the slope of Line 1 is \(-\frac{2}{3}\) and Line 2 is \(\frac{3}{2}\). When you multiply these slopes, the product is \(-1\), which proves they are perpendicular.
  • Perpendicular lines: Negative reciprocal slopes.
  • Intersect at a right angle.
  • Product of slopes is -1.
Line Equations
Line equations are used to represent lines on a Cartesian coordinate plane. They typically take the form of the slope-intercept equation: \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.

The y-intercept is where the line crosses the y-axis. For example, if a line has an equation \(y = 2x + 1\), it crosses the y-axis at \(y = 1\).

Knowing the slope and y-intercept, you can sketch the graph of the line quickly. This form is particularly favored because it allows easy reading of the slope, which indicates the line's steepness, and the y-intercept, which shows where it cuts through the y-axis.
  • Slope-intercept form: \(y = mx + b\).
  • Easy to identify slope and y-intercept.
  • Useful for graphing lines.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Sonya is walking home from a friend's house. After 2 minutes she is 1.4 miles from home. Twelve minutes after leaving, she is 0.9 miles from home. What is her rate?

A clothing business finds there is a linear relationship between the number of shirts, \(n,\) it can sell and the price, \(p,\) it can charge per shirt. In particular, historical data shows that 1000 shirts can be sold at a price of \(\$ 30\), while 3000 shirts can be sold at a price of \(\$ 22\). Find a linear equation in the form \(p=m n+b\) that gives the price \(p\) they can charge for \(n\) shirts.

Find the horizontal and vertical intercepts of each function. $$ f(x)=-2|x+1|+6 $$

Which of the following tables could represent a linear function? For each that could be linear, find a linear equation models the data. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{g}(\boldsymbol{x}) \\ \hline 0 & 5 \\ \hline 5 & -10 \\ \hline 10 & -25 \\ \hline 15 & -40 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{h}(\boldsymbol{x}) \\ \hline 0 & 5 \\ \hline 5 & 30 \\ \hline 10 & 105 \\ \hline 15 & 230 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{f}(\boldsymbol{x}) \\ \hline 0 & -5 \\ \hline 5 & 20 \\ \hline 10 & 45 \\ \hline 15 & 70 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{k}(\boldsymbol{x}) \\ \hline 5 & 13 \\ \hline 10 & 28 \\ \hline 20 & 58 \\ \hline 25 & 73 \\ \hline \end{array} $$

Determine if each function is increasing or decreasing $$ f(x)=4 x+3 $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.