/*! 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 51 For the following exercises, use... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 51

For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither? Line 1: Passes through (2,5) and (5,-1) Line 2: Passes through (-3,7) and (3,-5)

Short Answer

Expert verified
The lines are parallel since they have the same slope of -2.

Step by step solution

01

Identifying the Points

Line 1 passes through the points (2,5) and (5,-1). Line 2 passes through the points (-3,7) and (3,-5). We need these to find the slopes of both lines.
02

Calculating the Slope of Line 1

The slope of a line passing through two points (x鈧�, y鈧�) and (x鈧�, y鈧�) is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For Line 1, \((x_1, y_1) = (2, 5)\) and \((x_2, y_2) = (5, -1)\). Thus, the slope \( m_1 = \frac{-1 - 5}{5 - 2} = \frac{-6}{3} = -2 \).
03

Calculating the Slope of Line 2

Using the same slope formula, for Line 2 with points \((x_1, y_1) = (-3, 7)\) and \((x_2, y_2) = (3, -5)\), we find the slope \( m_2 = \frac{-5 - 7}{3 - (-3)} = \frac{-12}{6} = -2 \).
04

Comparing the Slopes

Two lines are parallel if they have the same slope and are perpendicular if the product of their slopes is -1. Here, the slopes of Line 1 and Line 2 are both -2. Therefore, both lines are parallel.

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 Calculation
Calculating the slope of a line is an essential skill in geometry. The slope provides a numerical value that represents the steepness and direction of a line.
A line's slope is calculated using the coordinates of two distinct points on the line. Here's the formula for slope calculation:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
  • The symbol \( m \) represents the slope.
  • \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
  • Subtract the y-coordinates and the x-coordinates separately to use in the formula.
Using these simple subtractions, we can determine the slope of Line 1 to be -2 and Line 2 also to be -2.
Intuitively, the slope tells us how much the line rises or falls as it moves from left to right. A negative slope means the line is descending.
Coordinate Geometry
Coordinate geometry, or analytic geometry, allows us to determine geometric relationships using algebra. It lets us use numerical equations to describe lines, curves, and shapes.
By assigning coordinates to points, the position and connection of these points are mathematically described. The coordinate system defines positions using ordered pairs, like (2,5) or (-3,7).
  • On a coordinate plane, the horizontal line is called the x-axis and the vertical line is the y-axis.
  • Coordinates are noted as \((x, y)\), describing a point's location relative to these axes.
In this context, we used the coordinates of two points on each line to compute the slope, demonstrating how geometry and algebra work together to solve spatial problems.
Line Equations
Line equations are foundational in coordinate geometry for defining straight lines. With a calculated slope and a point through which the line passes, we can write its equation in the slope-intercept form:\[y = mx + b\]
  • \( m \), the slope, is known from our calculations.
  • \( b \), the y-intercept, can be found by substituting one point's coordinates into the equation.
For example, with the slope of -2 for both lines, you can substitute \( m = -2 \) into the equation to find \( b \) using any point on the line.
Understanding line equations allows you to see how lines are expressed algebraically and how they behave on a graph. This knowledge helps determine relationships between lines, such as whether they are parallel or intersecting.

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

For the following exercises, consider this scenario: A town's population has been increased at a constant rate. In 2010 the population was \(4,020 .\) By 2012 the population had increased to \(52,070 .\) Assume this trend continues. Identify the year in which the population will reach \(75,000\) .

A phone company charges for service according to the formula: \(C(n)=24+0.1 n,\) where \(n\) is the number of minutes talked, and \(C(n)\) is the monthly charge, in dollars. Find and interpret the rate of change and initial value.

A phone company has a monthly cellular plan where a customer pays a flat monthly fee and then a certain amount of money per minute used on the phone. If a customer uses 410 minutes, the monthly cost will be \(\$ 71.50\) . If the customer uses 720 minutes, the monthly cost will be \(\$ 118\) . a. Find a linear equation for the monthly cost of the cell plan as a function of \(x,\) the number of monthly minutes used. b. Interpret the slope and \(y\) -intercept of the equation. c. Use your equation to find the total monthly cost if 687 minutes are used.

For the following exercises, consider this scenario: A town has an initial population of \(75,000 .\) It grows at a constant rate of \(2,500\) per year for 5 years. If the function \(P\) is graphed, find and interpret the \(x\) -and \(y\) -intercepts.

In 2003 , the owl population in a park was measured to be 340. By 2007, the population was measured again to be \(285 .\) The population changes linearly. Let the input be years since 1990 . a. Find a formula for the owl population, \(P .\) Let the input be years since 2003 . b. What does your model predict the owl population to be in 2012?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Applied 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.