/*! 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 37 Find the slope of each line. \... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 37

Find the slope of each line. \(2 x+y=7\)

Short Answer

Expert verified
The slope of the line is -2.

Step by step solution

01

Write the equation in slope-intercept form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Start by solving the given equation \( 2x + y = 7 \) for \( y \).
02

Isolate the y-variable

Subtract \( 2x \) from both sides of the equation to isolate the \( y \)-variable: \( y = -2x + 7 \).
03

Identify the slope

In the equation \( y = -2x + 7 \), the coefficient of \( x \) is the slope of the line. Thus, the slope \( m \) is \(-2\).

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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 one of the most straightforward ways to write the equation of a line. This form is given by the equation \( y = mx + b \), where:
  • \( m \) represents the slope of the line.
  • \( b \) is the y-intercept; it's the point where the line crosses the y-axis.
This form is particularly useful because it gives us direct access to the slope and the y-intercept, making it easier to graph the line and understand its steepness and direction.
When given any linear equation, such as \( 2x + y = 7 \), we often convert it into the slope-intercept form. This helps in quickly identifying the slope and the y-intercept, making the line equation more visually and analytically accessible.
Linear Equation
A linear equation is an equation that graphs a straight line on the coordinate plane. The standard form of a linear equation is given by \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. Here are a few key points about linear equations:
  • They do not involve any powers, roots, or products of variables.
  • They model constant rates of change.
Linear equations can represent many real-world situations, such as predicting costs, calculating speed, or understanding relationships between variables.
The initial equation in our exercise, \( 2x + y = 7 \), is a linear equation. By converting it to the slope-intercept form, we can easily find the slope and visualize the line it represents.
Isolate the Variable
To isolate a variable means to rearrange an equation so that one variable stands alone on one side of the equation. This process is key in solving equations and finding the slope of a line when dealing with linear equations.
For the equation \( 2x + y = 7 \), we isolate \( y \) to convert it into the slope-intercept form. Here鈥檚 how:
  • Subtract \( 2x \) from both sides to get \( y = -2x + 7 \).
By isolating \( y \), we effectively solve the equation for \( y \), allowing us to quickly identify that the slope \( m \) is \(-2\) from the new equation \( y = -2x + 7 \). Isolating the variable helps to make sense of the relationship between variables in mathematical terms and translates equations into a more understandable format.

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Most popular questions from this chapter

\(10+3 x=5 y\) \(5 x+3 y=1\)

Solve each equation for \(y.\) \(y-(-3)=4(x-(-5))\)

The average price of an acre of U.S. farmland was \(\$ 1132\) in \(2001 .\) In \(2006,\) the price of an acre rose to approximately \(\$ 1657 .\) (Source: National Agricultural Statistics Service) a. Write two ordered pairs of the form (year, price of acre) b. Find the slope of the line through the two points. c. Write a sentence explaining the meaning of the slope as a rate of change.

See Examples 1 through 7 . Find an equation of each line described. Write each equation in slope-intercept form (solved for \(y\) ), when possible. Slope \(-5, y\) -intercept \((0,7)\)

Show that a triangle with vertices at the points \((1,1),(-4,4)\) and \((-3,0)\) is a right triangle.
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