/*! 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 7 Find an equation of the given li... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 7

Find an equation of the given line. Slope is \(-1 ;(7,1)\) on line.

Short Answer

Expert verified
The equation of the line is: \( y = -x + 8 \)

Step by step solution

01

Formula for Slope-Intercept Form

Recall the slope-intercept form of a line: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept.
02

Substitute Slope into the Formula

Substitute the given slope \( m = -1 \) into the equation: \[ y = -1x + b \] This simplifies to: \[ y = -x + b \]
03

Use the Point to Find the Y-Intercept

To find the y-intercept \( b \), use the point (7, 1). Substitute \( x = 7 \) and \( y = 1 \) into the equation: \[ 1 = -7 + b \] Solve for \( b \): \[ b = 1 + 7 = 8 \]
04

Write the Final Equation

Substitute the value of \( b \) back into the equation: \[ y = -x + 8 \] So the equation of the line is: \[ y = -x + 8 \]

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.

equation of a line
To write the equation of a line, it's essential to use the slope-intercept form, which is \( y = mx + b \).
This formula gives us a direct way to describe a line when we know two key pieces of information: the slope (\( m \)) and the y-intercept (\( b \)).
The slope tells us how steep or flat the line is. It can be positive, negative, or even zero.
The y-intercept is where the line crosses (or intersects) the y-axis.
Once you know these two components, you can easily write the equation y = mx + b, making it clear and understandable.
finding y-intercept
Finding the y-intercept is a crucial step in creating the equation of a line.
In our example, we used the point (7, 1) to help find the y-intercept.
This step requires substituting the given point's coordinates into the partially completed equation and solving for \( b \).
In our case, we started with \( y = -x + b \),
which included the slope.
By replacing \( x = 7 \) and \( y = 1 \), we solved the equation:
  • 1 = -7 + b
  • 1 + 7 = b
  • b = 8

So the y-intercept is 8, meaning the line crosses the y-axis at this point.
substitute coordinates
Using coordinates to find missing parts in the equation is a fundamental concept.
To do this, follow these steps:
  • Identify the point's coordinates (\( x \) and \( y \)).
  • Substitute these values into your equation. In our example, \( y = -x + b \)
  • Replace \( x \) with the given x-coordinate and \( y \) with the given y-coordinate.
  • Solve for the unknown part, which is usually \( b \) (the y-intercept).
Substituting coordinates helps ensure that your final equation accurately represents the line through the given point.

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

The functions in Exercises 21-26 are defined for all \(x\) except for one value of \(x\). If possible, define \(f(x)\) at the exceptional point in a way that makes \(f(x)\) continuous for all \(x\). \(f(x)=\frac{(6+x)^{2}-36}{x}, x \neq 0\)

Apply the three-step method to compute the derivative of the given function. \(f(x)=3 x^{2}-2\)

In Exercises 33 and 34, deferm?ne the value of \(a\) that makes the function \(f(x)\) continuous at \(x=0\). \(f(x)=\left\\{\begin{array}{ll}1 & \text { for } x \geq 0 \\ x+a & \text { for } x<0\end{array}\right.\)

Each limit in Exercises 49-54 is a definition of \(f^{\prime}(a)\). Determine the function \(f(x)\) and the value of \(a\). \(\lim _{h \rightarrow 0} \frac{(1+h)^{-1 / 2}-1}{h}\)

For the given function, simultaneously graph the functions \(f(x), f^{\prime}(x)\), and \(f^{\prime \prime}(x)\) with the specified window setting. Note: Since we have not yet learned how to differentiate the given function, you must use your graphing utility's differentiation command to define the derivatives. $$f(x)=\frac{x}{1+x^{2}},[-4,4] \text { by }[-2,2]$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.