/*! 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 102 Graph each equation. \(\frac{x... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 102

Graph each equation. \(\frac{x}{2}-\frac{y}{3}-4=0\)

Short Answer

Expert verified
Graph by rewriting as \( y = \frac{3}{2}x - 12 \); slope \( m = \frac{3}{2} \) and y-intercept \( -12 \).

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

To graph the equation, we first need to express it in slope-intercept form, which is \( y = mx + b \). Start with the given equation: \( \frac{x}{2} - \frac{y}{3} - 4 = 0 \). Solve for \( y \) by first isolating \( \frac{y}{3} \) on one side. Add 4 to both sides to get \( \frac{x}{2} - \frac{y}{3} = 4 \). Next, subtract \( \frac{x}{2} \) from both sides to get \( -\frac{y}{3} = -\frac{x}{2} + 4 \). Multiply the entire equation by \(-3\) to eliminate the fraction: \( y = \frac{3}{2}x - 12 \). Now the equation is in the form \( y = mx + b \) with slope \( m = \frac{3}{2} \) and y-intercept \( b = -12 \).
02

Identify the Slope and Y-Intercept

From the equation \( y = \frac{3}{2}x - 12 \), identify the slope \( m \) as \( \frac{3}{2} \) which indicates how steep the line is, and the y-intercept \( b \) as \(-12\) where the line crosses the y-axis.
03

Plot the Y-Intercept

First, plot the y-intercept \(-12\) on the y-axis. This is the point \( (0, -12) \).
04

Use the Slope to Find Another Point

From the y-intercept \( (0, -12) \), use the slope \( \frac{3}{2} \) to find another point. The slope \( \frac{3}{2} \) means rise 3 units up and run 2 units to the right from the y-intercept. Moving from \( (0, -12) \) to \( (2, -9) \) provides another point on the line.
05

Draw the Line

Connect the two points, \( (0, -12) \) and \( (2, -9) \), with a straight line extending in both directions. This is the graph of the equation \( y = \frac{3}{2}x - 12 \).

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 one of the most commonly used formats for expressing linear equations. This form is particularly convenient for graphing because it clearly shows the slope and y-intercept of a line. The general form is expressed as:
\[ y = mx + b \]
Where:
  • \( m \) represents the slope of the line.
  • \( b \) stands for the y-intercept, the point where the line crosses the y-axis.
Understanding this form is crucial because it allows you to see at a glance whether a line is increasing or decreasing. Also, knowing the y-intercept helps you quickly find a starting point for graphing. By converting equations into this form, you can easily analyze and graph them.
Y-Intercept
The y-intercept is an important concept when dealing with linear equations in slope-intercept form. The y-intercept \( b \) is the value at which the graph of the equation crosses the y-axis. This occurs when the coordinate for \( x \) is zero.
For example, in the equation \( y = \frac{3}{2}x - 12 \), the y-intercept is \(-12\). This tells us that the line will intersect the y-axis at the point (0, -12).
To graph a line using its y-intercept:
  • Start by plotting the y-intercept on the y-axis. This provides a starting point for the line.
  • Use the slope to determine the direction and steepness of the line from this point.
The y-intercept provides a handy reference point that simplifies the graphing process and helps visualize the equation.
Linear Equations
Linear equations are equations that, when graphed, produce a straight line. They have consistent rates of change and are often represented in the format of \( ax + by = c \). When solving and graphing a linear equation, converting it to slope-intercept form \( y = mx + b \) can make the task simpler.
Key traits of linear equations include:
  • A constant slope that remains the same between any two points along the line.
  • The line continues infinitely in both directions, depicting an endless, straight path.
Such equations are foundational in algebra because they model many real-life situations like speed, economics, and growth patterns. Understanding linear equations is essential for solving problems related to these areas, and they often serve as a stepping stone for more complex mathematical concepts.

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

a. Suppose you know the slope of a line. Is that enough information about the line to write its equation? Explain. b. Suppose you know the coordinates of a point on a line. Is that enough information about the line to write its equation? Explain.

Write an equation for a linear function whose graph has the given characteristics. Slope \(\frac{1}{4},\) passes through \((8,1)\)

Write an equation for a linear function whose graph has the given characteristics. Passes through \((1,2),\) perpendicular to the graph of \(g(x)=-\frac{x}{6}+1\)

Write an equation for a linear function whose graph has the given characteristics. Slope \(\frac{1}{5},\) passes through \((10,1)\)

The function $$T(a)=837.50+0.15(a-8,375)$$ (where \(a\) is adjusted gross income) is a model of the instructions given on the first line of the following tax rate Schedule X. a. Find \(T(25,000)\) and interpret the result. b. Write a function that models the second line on Schedule X.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Pure 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.