/*! 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 272 Find the equation of a line with... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 272

Find the equation of a line with a \(y\) - intercept of \((0,2)\) and slope \(-\frac{1}{2}\).

Short Answer

Expert verified
The equation of the line is \(y = -\frac{1}{2}x + 2\).

Step by step solution

01

Identify the known variables

The slope of the line is given as \(-\frac{1}{2}\), and the y-intercept is given as the point \((0,2)\). These values will help us form the equation of the line.
02

Understand the slope-intercept form of a line

The slope-intercept form of a line is given by the equation \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.
03

Substitute the known values into the slope-intercept form

Replace \(m\) in \(y = mx + c\) with the slope \(-\frac{1}{2}\) and \(c\) with the y-intercept 2. The equation becomes:\[ y = -\frac{1}{2}x + 2 \]
04

Confirm the equation

Verify that the equation \(y = -\frac{1}{2}x + 2\) correctly represents a line with the given slope and y-intercept by checking that when \(x = 0\), \(y = 2\). This confirms the y-intercept and when plotted, a slope of -\(\frac{1}{2}\) means for every increase of 1 in \(x\), \(y\) decreases by \(\frac{1}{2}\).

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
The slope-intercept form of a linear equation is a fundamental concept in algebra. It provides a straightforward way to write the equation of a line and understand its properties. The form is represented as:
  • \( y = mx + c \)
Here:
  • \( y \) represents the dependent variable (usually the vertical coordinate in a graph).
  • \( x \) is the independent variable (usually the horizontal coordinate).
  • \( m \) is the slope of the line, showing the rate of change of \( y \) with respect to \( x \).
  • \( c \) is the y-intercept, the point where the line crosses the y-axis.
This form is particularly useful because it allows us to quickly identify the slope and y-intercept, making calculations and graphing much simpler. When given values for \( m \) and \( c \), you can easily substitute them into the equation to describe the specific line.
Y-Intercept
The y-intercept is a crucial feature of the slope-intercept form. It tells us where the line meets the y-axis, providing a starting point for the graph of the line. Mathematically, the y-intercept occurs when \( x = 0 \). In our example, the y-intercept is 2, which is represented by the point \( (0, 2) \).

This means that the line crosses the y-axis at 2 on the graph. It's a direct indication of the line's vertical position on the coordinate plane when \( x \) has no effect. The y-intercept gives valuable information about the line's overall position and helps in sketching its graph precisely.
  • A positive y-intercept moves the line above the origin.
  • A negative y-intercept positions the line below the origin.
Slope of a Line
The slope of a line measures its steepness and direction and is one of the most important aspects of linear equations. It is calculated as the ratio of the change in \( y \) (vertical change) to the change in \( x \) (horizontal change). The formula for slope \( m \) is:
  • \( m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \)
The slope tells us how much \( y \) changes for a one-unit change in \( x \):
  • A positive slope indicates that as \( x \) increases, \( y \) also increases, creating an upward slant.
  • A negative slope, like \(-\frac{1}{2}\), means that as \( x \) increases, \( y \) decreases, giving a downward tilt.
  • A zero slope signifies a horizontal line with constant \( y \) values.
Understanding the slope is key to analyzing the behavior of the line and predicting points that lie on it. It reflects how quickly or slowly the line increases or decreases across the coordinate plane.

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, use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither? Line \(1 :\) Passes through \((5,11)\) and \((10,1)\) Line \(2 :\) Passes through \((-1,3)\) and \((-5,11)\)

A city's population in the year 1960 was \(287,500 .\) In 1989 the population was \(275,900 .\) Compute the rate of growth of the population and make a statement about the population rate of change in people per year.

Write an equation for a line perpendicular to \(f(x)=5 x-1\) and passing through the point \((5,20)\).

Determine whether the following function is increasing or decreasing. \(f(x)=-2 x+5\)

For the following exercises, consider the data in Table 2.18, which shows the percent of unemployed in a city of people 25 years or older who are college graduates is given below, by year. $$\begin{array}{|c|c|c|c|c|c|}\hline \text { Year } & {2000} & {2002} & {2005} & {2007} & {2010} \\ \hline \text { Percent Graduates } & {6.5} & {7.0} & {7.4} & {8.2} & {9.0} \\ \hline\end{array}$$ Determine whether the trend appears to be linear. If so, and assuming the trend continues, find a linear regression model to predict the percent of unemployed in a given year to three decimal places
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Mechanics 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.