/*! 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 41 Determine the slope of the line ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 41

Determine the slope of the line from its equation. $$x+y=7$$

Short Answer

Expert verified
The slope of the line is \(-1\).

Step by step solution

01

Rewrite the Equation

Rewrite the given equation in the slope-intercept form, which is written as \[ y = mx + b \]here, \(m\) represents the slope and \(b\) represents the y-intercept. The given equation is \[ x + y = 7 \]First, solve for \(y\).
02

Isolate the Variable y

Subtract \(x\) from both sides of the equation to isolate \(y\):\[ x + y - x = 7 - x \]which simplifies to \[ y = -x + 7 \]
03

Identify the Slope

Now that the equation is in the slope-intercept form \[ y = -x + 7 \]identify the slope \(m\) as the coefficient of \(x\). In this case, the coefficient of \(x\) is \(-1\). So, the slope is \(-1\).

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
When working with linear equations, it's incredibly useful to write them in the slope-intercept form. This form is expressed as \( y = mx + b \). Here, \( m \) stands for the slope of the line, and \( b \) represents the y-intercept, where the line crosses the y-axis. By rearranging equations into this form, it becomes easy to immediately see both the slope and y-intercept. For example, in the given problem, the equation \( x + y = 7 \) was rearranged to \( y = -x + 7 \) to identify the slope and y-intercept quickly.
linear equations
Linear equations represent relationships where the dependent variable changes at a constant rate with the independent variable. The general form of a linear equation in two variables is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. These equations graph as straight lines. To make working with them simpler, they can often be rewritten into the slope-intercept form. This transformation lets you easily spot the slope and the y-intercept. For instance, from the equation \( x + y = 7 \), we can rewrite it to \( y = -x + 7 \), making it clear that the slope of the line is \(-1\) and its y-intercept is \(7\).
solving for y
Solving for \( y \) in an equation is an important step in converting any linear equation to the slope-intercept form. It involves isolating \( y \) on one side of the equation. Let's look at the example provided. The equation \( x + y = 7 \) needs to be rearranged to solve for \( y \).
  • Start by subtracting \( x \) from both sides: \( x + y - x = 7 - x \)
  • This simplifies to \( y = -x + 7 \).
Now the equation is in the form \( y = mx + b \), making it straightforward to identify the slope and y-intercept. For this problem, the slope \( m \) is \(-1\) and the y-intercept \( b \) is \(7\).

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

Write an equation of the line that passes through the point \((-3,0)\) and is parallel to the line whose equation is \(y=-x+4\)

Determine the slope of the line from its equation. $$y=-3 x-1$$

Write an equation of the line satisfying the given conditions. Line has \(x\) -intercept \(-3\) and \(y\) -intercept 4

A slanted roof on a house rises 2.8 meters over a horizontal distance of 6.2 meters. Find the grade (or pitch) of the roof.

Sketch the graph of the line satisfying the given conditions. Passing through \((-1,0)\) with slope 4
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

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.