/*! 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 31 For each equation, find the slop... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 31

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. $$ \frac{2 x}{3}-y=1 $$

Short Answer

Expert verified
Slope \( m = \frac{2}{3} \), y-intercept \((0, -1)\).

Step by step solution

01

Rewrite in Slope-Intercept Form

Start by rewriting the given equation \( \frac{2x}{3} - y = 1 \) into the standard slope-intercept form \( y = mx + b \). To do this, solve for \( y \):\[ \frac{2x}{3} - y = 1 \]. Add \( y \) to both sides to get: \[ \frac{2x}{3} = y + 1 \]. Next, subtract 1 from both sides: \[ \frac{2x}{3} - 1 = y \]. Finally, rewrite it in the form \( y = mx + b \): \( y = \frac{2}{3}x - 1 \).
02

Identify the Slope and Y-Intercept

The equation \( y = \frac{2}{3}x - 1 \) is now in the standard form \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. Thus, the slope \( m \) is \( \frac{2}{3} \), and the \( y \)-intercept \( b \) is \( -1 \). Therefore, the point for the \( y \)-intercept is \((0, -1)\).
03

Draw the Graph

To draw the graph of the line, start by plotting the \( y \)-intercept, which is the point \((0, -1)\) on the y-axis. Next, use the slope \( \frac{2}{3} \) to plot the next point. The slope indicates that for every 3 units moved to the right (in the positive \( x \)-direction), the graph rises by 2 units (in the positive \( y \)-direction). From the point \((0, -1)\), move 3 units to the right to \((3, -1)\) and then 2 units up to \((3, 1)\). Draw a straight line through these points to get the graph of the line.

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.

Graphing Linear Equations
When graphing linear equations, the goal is to draw a line that represents all the solutions of the equation. A linear equation is an equation that forms a straight line when graphed, and it can be represented in various forms, with the slope-intercept form being the most common. To begin graphing, start by identifying at least two points on the line, as a straight line is defined by just two points.

Here鈥檚 how:
  • Find the y-intercept, the point where the line crosses the y-axis.
  • Use the slope, which is the ratio of the change in y to the change in x, to determine another point.
Once you have these points, draw a straight line through them, extending in both directions. Remember, the line continues infinitely, so be sure to stretch it across the graph. This visual representation helps understand the relationship described by the equation.
Finding Slope and Y-Intercept
To master graphing, it's crucial to understand how to find the slope and y-intercept of a linear equation. The slope, represented as \( m \), indicates how steep the line is. The y-intercept, denoted by \( b \), is where the line crosses the y-axis.

Follow these steps:
  • The slope \( m \) can be understood as "rise over run." It tells you how many units the line goes up or down for every unit it goes horizontally to the right.
  • The y-intercept \( b \) is simply the y-coordinate of the point where the line meets the y-axis.
In the equation \( y = \frac{2}{3}x - 1 \), the slope \( m \) is \( \frac{2}{3} \), meaning the line goes up 2 units for every 3 units you move to the right. The y-intercept \( b \) is \(-1\), telling us that the line crosses the y-axis at the point \((0, -1)\). This understanding helps predict the behavior of the line.
Rewriting Equations
Rewriting equations into different forms can make them easier to work with, especially when graphing. The slope-intercept form of a linear equation is particularly useful because it directly shows the slope \( m \) and y-intercept \( b \).

Here鈥檚 how you can transform an equation into this form:
  • Start with the original equation, such as \( \frac{2x}{3} - y = 1 \).
  • Rearrange it to solve for \( y \). This involves moving \( y \) to one side and the other terms to the opposite side.
  • In our example, by adding \( y \) and then subtracting 1, you get \( y = \frac{2}{3}x - 1 \).
This form quickly reveals the slope and y-intercept, making graphing straightforward. Rewriting equations can simplify problem-solving and provide greater insight into the relationships expressed by the equation.

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

101-102. GENERAL: Waterfalls Water falling from a waterfall that is \(x\) feet high will hit the ground with speed \(\frac{60}{11} x^{0.5}\) miles per hour (neglecting air resistance). Find the speed of the water at the bottom of the highest waterfall in the world, Angel Falls in Venezuela ( 3281 feet high).

Simplify. $$ \frac{\left(w w^{2}\right)^{3}}{w^{3} w} $$

When defining \(x^{m / n}\), why did we require that the exponent \(\frac{\mathrm{m}}{\mathrm{u}}\) be fully reduced? [Hint: \((-1)^{2 / 3}=(\sqrt[3]{-1})^{2}=1,\) but with an equal but unreduced exponent you get \((-1)^{4 / 6}=(\sqrt[6]{-1})^{4}\). Is this defined?]

ATHLETICS: Cardiovascular Zone Your maximum heart rate (in beats per minute) may be estimated as 220 minus your age. For maximum cardiovascular effect, many trainers recommend raising your heart rate to between \(50 \%\) and \(70 \%\) of this maximum rate (called the cardio zone). a. Write a linear function to represent this upper limit as a function of \(x\), your age. Then write a similar linear function to represent the lower limit. Use decimals instead of percents. b. Use your functions to find the upper and lower cardio limits for a 20-year- old person. Find the cardio limits for a 60-year-old person.

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

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.