/*! 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 5 Lines with the same slope: On th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 5

Lines with the same slope: On the same coordinate axes, draw two lines, each of slope 2 . The first line has vertical intercept 1 , and the second has vertical intercept 3. Do the lines cross? In general, what can you say about different lines with the same slope?

Short Answer

Expert verified
The lines do not cross. Lines with the same slope are parallel.

Step by step solution

01

Understanding the Line Equation

The equation of a line in slope-intercept form is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept (vertical intercept). In this problem, both lines have a slope \( m = 2 \).
02

Write Equations for Both Lines

The first line has a slope of 2 and a vertical intercept of 1. Thus, its equation is \( y = 2x + 1 \). The second line also has a slope of 2 but a vertical intercept of 3, so its equation is \( y = 2x + 3 \).
03

Analyze Intersection

For two lines to intersect, their equations must have a common solution \((x, y)\). Set the equations equal to each other: \( 2x + 1 = 2x + 3 \). Solving this equation, we get \( 1 = 3 \), which is a contradiction, indicating that the lines do not intersect.
04

Conclusion About Parallel Lines

Lines with the same slope are parallel; thus they never intersect unless they have the same intercept, in which case they are the same line. Therefore, lines with the same slope and different intercepts will not cross.

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.

Slope-Intercept Form
When discussing the equation of a line, understanding the slope-intercept form is crucial. This form is expressed as \( y = mx + b \), where:
  • \( y \) and \( x \) are variables representing the coordinates of any point on the line.
  • \( m \) represents the slope of the line. This value tells us how steep the line is, determining the rise over run or the change in \( y \) for each unit change in \( x \).
  • \( b \) is the y-intercept. It defines where the line crosses the y-axis, indicating the value of \( y \) when \( x = 0 \).
The beauty of the slope-intercept form is its simplicity and clarity in revealing the characteristics of a line. By using \( y = 2x + 1 \) as an example, you can immediately identify that the slope is 2, meaning the line rises 2 units for every 1 unit it moves horizontally. The y-intercept is 1, showing where it intersects the y-axis.
Equation of a Line
An equation of a line can tell you many things about its orientation and location on a graph. When we write a line in the form \( y = mx + b \), we're defining it with precision.
  • The slope \( m \) controls the tilt of the line. A positive value indicates an upward trend, while a negative value suggests it's going downward.
  • The y-intercept \( b \) shows precisely where the line will meet the y-axis.
For instance, in the example equation \( y = 2x + 3 \), the slope \( m = 2 \) means the line will rise twice as fast as it moves across. Additionally, the y-intercept \( b = 3 \) tells us the line crosses the y-axis at the point (0, 3). This clarity makes solving and graphing linear equations much easier. Remember, lines with the same slope are parallel, and understanding the equation helps identify this property.
Y-Intercept
The y-intercept is a key concept when exploring lines and their graphs. It is simply the point where a line crosses the y-axis. In terms of the equation \( y = mx + b \), \( b \) is the y-intercept. Its role is quite straightforward:
  • It provides a starting point on the y-axis when graphing a line.
  • Being the constant in the equation, it shows the value of \( y \) when \( x = 0 \).
For example, in the line equation \( y = 2x + 1 \), the y-intercept is at 1. It dictates that when \( x \) is zero, the value of \( y \) will be 1. Having different y-intercepts, as in \( y = 2x + 1 \) and \( y = 2x + 3 \), shows us that the lines, though parallel, start at different points on the y-axis. This distinction is essential to identify parallel lines and evaluate their relationship on a graph.

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

Competition between populations: In this exercise we consider the problem of competition between two populations that vie for resources but do not prey on each other. Let \(m\) be the size of the first population, let \(n\) be the size of the second (both measured in thousands of animals), and assume that the populations coexist eventually. An example of one common model for the interaction is Per capita growth rate for \(m\) is \(3(1-m-n)\) Per capita growth rate for \(n\) is $$ 2(1-0.7 m-1.1 n) $$ At an equilibrium point the per capita growth rates for \(m\) and for \(n\) are both zero. If the populations reach such a point, then they will continue at that size indefinitely. Find the equilibrium point in the example above.

An application of three equations in three unknowns: A bag of coins contains nickels, dimes, and quarters. There are a total of 21 coins in the bag, and the total amount of money in the bag is \(\$ 3.35\). There is one more dime than there are nickels. How many dimes, nickels, and quarters are in the bag?

Getting Celsius from Fahrenheit: Water freezes at 0 degrees Celsius, which is the same as 32 degrees Fahrenheit. Also water boils at 100 degrees Celsius, which is the same as 212 degrees Fahrenheit. a. Use the freezing and boiling points of water to find a formula expressing Celsius temperature \(C\) as a linear function of the Fahrenheit temperature \(F\). b. What is the slope of the function you found in part a? Explain its meaning in practical terms. c. In Example 3.5 we showed that \(F=1.8 C+32\). Solve this equation for \(C\) and compare the answer with that obtained in part a.

Another interesting system of equations: What happens when you try to solve the following system of equations? Can you explain what is going on? $$ \begin{gathered} x+2 y=3 \\ -2 x-4 y=-6 \end{gathered} $$

Measuring the circumference of the Earth: Eratosthenes, who lived in Alexandria around 200 B.C., learned that at noon on the summer solstice the sun shined vertically into a well in Syene (modern Aswan, due south of Alexandria). He found that on the same day in Alexandria the sun was about 7 degrees short of being directly overhead. Since 7 degrees is about \(\frac{1}{50}\) of a full circle of 360 degrees, he concluded that the distance from Syene to Alexandria was about \(\frac{1}{50}\) of the Earth's circumference. He knew from travelers that it was a 50-day trip and that camels could travel 100 stades \(^{13}\) per day. What was Eratosthenes' measure, in stades, of the circumference of the Earth? One estimate is that the stade is \(0.104\) mile. Using this estimate, what was Eratosthenes' measurement of the circumference of the Earth in miles?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

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.