/*! 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 29 Find an equation of the line: ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 29

Find an equation of the line: with \(m=-2\), containing (2,3).

Short Answer

Expert verified
The equation of the line is \( y = -2x + 7 \).

Step by step solution

01

Understanding the Problem

To find the equation of a line, we need to use the slope-intercept form, which is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We are given that \( m = -2 \) and the line contains the point \((2, 3)\).
02

Substitute the Slope

Substitute the value of the slope \( m = -2 \) into the slope-intercept form equation. This gives us \( y = -2x + b \).
03

Substitute the Point

The line passes through the point \( (2, 3) \). Substitute \( x = 2 \) and \( y = 3 \) into the equation \( y = -2x + b \) to find \( b \): \( 3 = -2(2) + b \).
04

Solve for the Y-intercept

Simplify the equation \( 3 = -4 + b \) to solve for \( b \). This gives \( b = 3 + 4 \), so \( b = 7 \).
05

Write the Equation

Now substitute \( b = 7 \) back into the equation \( y = -2x + b \). The equation of the line is \( y = -2x + 7 \).

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 is a common way to express the equation of a straight line. It's written as \( y = mx + b \). This form is particularly useful because it clearly displays two essential characteristics of the line:
  • The slope \( m \): Indicates how steep the line is.
  • The y-intercept \( b \): Tells us where the line crosses the y-axis.
To derive an equation using this form, simply plug in the slope of the line and solve for the y-intercept. This makes finding the equation of a line straightforward when both the slope and a point on the line are known.
Slope of a Line
The slope of a line, represented by \( m \), is a measure of its steepness. Mathematically, slope is calculated as the "rise" over the "run," or the change in \( y \) over the change in \( x \). In symbols, this is \( m = \frac{\Delta y}{\Delta x} \).
  • A positive slope means the line inclines upwards from left to right.
  • A negative slope indicates it declines.
  • Zero slope signifies a horizontal line.
In our example, the slope \( m \) is \(-2\). This means for every unit increase in \( x \), \( y \) decreases by 2 units. This negative slope tells us the line will slant downwards.
Y-Intercept
The y-intercept, denoted as \( b \), is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), \( b \) is the constant term in the equation. It signifies the \( y \)-value when \( x \) is zero.
In the context of our problem, after substituting the slope and the given point into the equation, we found that \( b = 7 \). This means the line crosses the y-axis at the point \((0, 7)\).
Understanding the y-intercept is crucial because it provides a starting position of the line on the graph. Knowing it allows you to visualize where the line begins before adjusting further based on the slope.

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

$$ \begin{aligned} &\text { A function } f \text { is given by }\\\ &f(x)=|x-2|+|x+1|-5\\\ &\text { Find } f(-3), f(-2), f(0), \text { and } f(4) \end{aligned} $$

Find the equilibrium point for each pair of demand and supply functions. Demand: \(q=8800-30 x ; \quad\) Supply: \(q=7000+15 x\)

The Technology Connection heading indicates exercises designed to provide practice using a graphing calculator. Graph. \(y=x^{3}+2 x^{2}-4 x-13\)

The number of tickets sold for a fund-raiser is inversely proportional to the price of a ticket, \(p\). If 175 tickets can be sold for \(\$ 20\) each, how many tickets will be sold if the price is \(\$ 25\) each?

Use the ZERO feature or the INTERSECT feature to approximate the zeros of each function to three decimal places. $$ \begin{aligned} f(x)=& x^{8}+8 x^{7}-28 x^{6}-56 x^{5}+70 x^{4}+56 x^{3} \\ &-28 x^{2}-8 x+1 \end{aligned} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

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.