/*! 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 80 Find the slope and the \(y\) -in... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 80

Find the slope and the \(y\) -intercept of the line with equation \(2 x-9 y+1=0\)

Short Answer

Expert verified
Slope: \( \frac{2}{9} \). Y-intercept: \( \frac{1}{9} \).

Step by step solution

01

Rearrange the equation into slope-intercept form

The slope-intercept form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Start by isolating \( y \) on one side of the equation.Given equation: \[ 2x - 9y + 1 = 0 \]Subtract \( 2x \) and \( 1 \) from both sides to isolate \( -9y \):\[ -9y = -2x - 1 \]
02

Solve for y

Divide every term by \( -9 \) to solve for \( y \):\[ y = \frac{2}{9} x + \frac{1}{9} \]Now the equation is in slope-intercept form, \( y = mx + b \).
03

Identify the slope and y-intercept

From the slope-intercept form of the equation \( y = \frac{2}{9} x + \frac{1}{9} \), identify the slope \( m \) and the y-intercept \( b \):Slope \( m = \frac{2}{9} \)Y-intercept \( b = \frac{1}{9} \)

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 commonly used way to express linear equations. The standard form is \ \( y = mx + b \ \). In this equation, \( m \ \) represents the slope of the line, and \( b \ \) represents the y-intercept. This form is very useful because it provides a clear and straightforward way to identify both the slope and the y-intercept directly from the equation.
When working with linear equations, converting them to slope-intercept form can make many tasks, such as graphing, much simpler.
You just need to isolate \( y \ \) on one side of the equation, as shown in the solution above.
linear equation
A linear equation is an equation that represents a straight line when graphed on a coordinate plane. These equations can be written in several forms, including slope-intercept form, standard form, and point-slope form.
In any linear equation, each term is either a constant or the product of a constant and a single variable. The general form of a linear equation is \( Ax + By + C = 0 \ \), where \( A \ \), \( B \ \), and \( C \ \) are constants.
As demonstrated in the example problem, you can rearrange a linear equation to isolate \( y \ \) and rewrite it in slope-intercept form. This makes it easy to identify the slope and y-intercept.
rearranging equations
Rearranging equations is a key skill in algebra that involves manipulating the equation to isolate a specific variable. This process often requires adding, subtracting, multiplying, or dividing terms on both sides of the equation.
In our example, we needed to rearrange the given equation \( 2x - 9y + 1 = 0 \ \) to isolate \( y \ \). We did this by:
  • Subtracting \( 2x \ \) and \( 1 \ \) from both sides
  • Then, dividing every term by \( -9 \ \) to solve for \( y \ \)
This step is critical as it transforms the equation into a more usable form for identifying specific properties like the slope and y-intercept.
slope
The slope of a line measures its steepness and direction. It is denoted by the variable \( m \ \) in the slope-intercept form \( y = mx + b \ \).
Mathematically, the slope is calculated as the ratio of the change in \( y \ \) (vertical change) to the change in \( x \ \) (horizontal change) between two points on the line, represented as \( m = \frac{\Delta y}{\Delta x} \ \).
In our example, once we rearranged the equation into slope-intercept form, we identified the slope \( m \ \) as \( \frac{2}{9} \ \), which means for every 9 units moved horizontally, the line rises by 2 units.
y-intercept
The y-intercept of a line is the point where it crosses the y-axis. This is represented by the variable \( b \ \) in the slope-intercept form \( y = mx + b \ \).
The y-intercept occurs when the value of \( x \ \) is 0, so you are left with \( y = b \ \).
In our example, after rearranging the equation into slope-intercept form, the y-intercept \( b \ \) is identified as \( \frac{1}{9} \ \). This means the line crosses the y-axis at the point \( (0, \frac{1}{9}) \ \).
This is crucial for graphing linear equations as it gives a starting point for drawing the line.

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 equations of two functions \(f\) and \(g\) such that \(f \circ g=g \circ f=x .\) (In Section 5.1, we will study inverse functions. If \(f \circ g=g \circ f=x,\) functions \(f\) and \(g\) are inverses of each other.)

Find \(f(x)\) and \(g(x)\) such that \(h(x)=(f \circ g)(x)\) Answers may vary. $$h(x)=\left|9 x^{2}-4\right|$$

Find an equation of variation for the given situation. \(y\) varies jointly as \(x\) and the square of \(z,\) and \(y=105\) when \(x=14\) and \(z=5\).

Find \(f(x)\) and \(g(x)\) such that \(h(x)=(f \circ g)(x)\) Answers may vary. $$h(x)=(x+2)^{3}-5(x+2)^{2}+3(x+2)-1$$

Let \(p(a)\) represent the number of pounds of grass seed required to seed a lawn with area \(a .\) Let \(c(s)\) represent the cost of \(s\) pounds of grass seed. Which composition makes sense: \((c \circ p)(a)\) or \((p \circ c)(s) ?\) What does it represent?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

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.