/*! 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 36 Write an equation of the line sa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 36

Write an equation of the line satisfying the following conditions. If possible, write your answer in the form \(y=m x+b\). Slope -1 and passing through the point (4,3)

Short Answer

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

Step by step solution

01

Identify the slope-intercept form equation

The slope-intercept form of a linear equation is given by the formula \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
02

Use the given slope

We are given the slope \(m = -1\). Substitute \(m = -1\) into the equation. Now the equation is \(y = -x + b\).
03

Substitute the point into the equation

We need to find \(b\), the y-intercept. Use the point \((4, 3)\), meaning \(x = 4\) and \(y = 3\). Substitute these values into the equation \(y = -x + b\): \(3 = -(4) + b\).
04

Solve for b

Simplify the equation: \(3 = -4 + b\). Add 4 to both sides to solve for \(b\): \(3 + 4 = b\). Thus, \(b = 7\).
05

Write the final equation

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

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 fundamental representation of a linear equation. It is written as \( y = mx + b \), where:
  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
This form is particularly useful because it allows us to easily graph a line and understand its behavior with just the slope and the y-intercept. For any linear equation, if you know these two values, you can describe the line completely. The formula provides a straightforward way to see what happens to \( y \) as \( x \) changes.
Slope
The slope of a line, denoted by \( m \), is a measure of its steepness. It tells us how much \( y \) increases or decreases as \( x \) increases by 1 unit. The formula to find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]The slope can be positive, negative, zero, or undefined:
  • Positive slope: line rises from left to right.
  • Negative slope: line falls from left to right.
  • Zero slope: horizontal line.
  • Undefined slope: vertical line.
In the example, we have a slope of \(-1\), which indicates the line decreases one unit in \(y\) for each unit increase in \(x\).
Y-Intercept
The y-intercept of a line is the value of \( y \) when \( x = 0 \). It is represented by \( b \) in the slope-intercept form \( y = mx + b \). In practical terms, it is the point where the line crosses the y-axis:- The y-intercept provides a starting point for graphing the line.In our example, after calculating using the point \((4, 3)\), the y-intercept \( b \) was found to be 7. Thus, the line crosses the y-axis at \( (0, 7) \). This means if you start at the origin, the line will pass through the point on the y-axis at 7.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, involves studying geometry using a coordinate system. It helps us understand concepts of lines and curves algebraically.
  • Its foundation lies in the coordinate plane which consists of two axes: x-axis (horizontal) and y-axis (vertical).
  • Each point on the plane is represented as \((x, y)\), a pair of numerical values showing its direction and distance from the origin \((0, 0)\).
In coordinate geometry, the relationship between coordinates is expressed through equations. For example, the equation of a line \( y = mx + b \) establishes a straight line by showing how \( y \) changes linearly with \( x \). In this exercise, the given point \((4, 3)\) helps determine the y-intercept when the slope is known. By substituting the point into the line's equation, we anchor the line within 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

BUSINESS: Semiconductor Sales The following table shows worldwide sales for semiconductors used in cell phones and laptop computers for recent years. \begin{tabular}{lccc} \hline Year & 2011 & 2012 & 2013 \\ \hline Sales (billions \$) & 80.2 & 87.1 & 93.6 \\ \hline \end{tabular} a. Number the data columns with \(x\) -values \(1-3\) (so that \(x\) stands for years since 2010 ), use power regression to fit a power curve to the data, and state the regression formula. [Hint: See Example 7.] b. Use the regression formula to predict sales in \(2020 .\) [Hint: What \(x\) -value corresponds to \(2020 ?]\)

For each statement, state in words the values of \(x\) for which each exponential expression is defined. $$ x^{1 / 3} $$

\(97-98 .\) GENERAL: Earthquakes The sizes of major earthquakes are measured on the Moment Magnitude Scale, or MMS, although the media often still refer to the outdated Richter scale. The MMS measures the total energy relensed by an earthquake, in units denoted \(M_{W}\) (W for the work accomplished). An increase of \(1 M_{W}\) means the energy increased by a factor of \(32,\) so an increase from \(A\) to \(B\) means the energy increased by a factor of \(32^{B-A}\). Use this formula to find the increase in energy between the following earthquakes: The 1994 Northridge, California, earthquake that measured \(6.7 \mathrm{M}_{\mathrm{W}}\) and the 1906 San Francisco earthquake that measured \(7.8 \mathrm{M}_{W}\). (The San Francisco earthquake resulted in 3000 deaths and a 3 -day fire that destroyed 4 square miles of San Francisco.)

ALLOMETRY: Dinosaurs The study of size and shape is called "allometry," and many allometric relationships involve exponents that are fractions or decimals. For example, the body measurements of most four-legged animals, from mice to elephants, obey (approximately) the following power law: $$ \left(\begin{array}{c} \text { Average body } \\ \text { thickness } \end{array}\right)=0.4 \text { (hip-to-shoulder length) }^{3 / 2} $$ where body thickness is measured vertically and all measurements are in feet. Assuming that this same relationship held for dinosaurs, find the average body thickness of the following dinosaurs, whose hip-toshoulder length can be measured from their skeletons: Triceratops, whose hip-to-shoulder length was 14 feet.

If an object is thrown upward so that its height (in feet) above the ground \(t\) seconds after it is thrown is given by the function \(h(t)\) below, find when the object hits the ground. That is, find the positive value of \(t\) such that \(h(t)=0 .\) Give the answer correct to two decimal places. [Hint: Enter the function in terms of \(x\) rather than \(t .\) Use the ZERO operation, or TRACE and ZOOM IN, or similar operations. $$ h(t)=-16 t^{2}+40 t+4 $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Applied 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.