/*! 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 30 For each equation, find the slop... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 30

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

Short Answer

Expert verified
The slope is \(-\frac{3}{2}\) and the y-intercept is 3.

Step by step solution

01

Rearrange the Equation

The given equation is \( \frac{x}{2} + \frac{y}{3} = 1 \). Our goal is to rearrange it into the slope-intercept form, \( y = mx + b \).Start by isolating \( y \):\[ \frac{y}{3} = 1 - \frac{x}{2} \]Now, multiply each term by 3 to clear the fraction:\[ y = 3 - \frac{3}{2}x \]
02

Identify the Slope and Intercept

The equation \( y = 3 - \frac{3}{2}x \) is in the slope-intercept form \( y = mx + b \).Here, the slope \( m \) is \( -\frac{3}{2} \) and the y-intercept \( b \) is 3. So the y-intercept point is \((0, 3)\).
03

Plot the Y-Intercept

Begin the graph by marking the y-intercept point \((0, 3)\) on the coordinate plane.
04

Use the Slope to Find Another Point

The slope \( m = -\frac{3}{2} \) indicates that for every 2 units you move to the right along the x-axis, you move 3 units down (since the slope is negative) along the y-axis.From the y-intercept \((0, 3)\), move 2 units to the right to \((2, 3)\), then 3 units down to \((2, 0)\). Mark the point \((2, 0)\) on the graph.
05

Draw the Line

Draw a straight line through the points \((0, 3)\) and \((2, 0)\). This line is the graph of the equation \(\frac{x}{2} + \frac{y}{3} = 1\).

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.

Coordinate Plane
The coordinate plane is a two-dimensional surface formed by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin, which has the coordinates \((0, 0)\).
Each point on the plane can be identified with a unique ordered pair \((x, y)\), where \(x\) represents the horizontal position and \(y\) represents the vertical position.
Important elements:
  • The x-axis helps you locate a point horizontally, while the y-axis assists in finding a location vertically.
  • Points to the right of the origin have positive x-values, and points above the origin have positive y-values.
  • Points to the left of the origin have negative x-values, and points below the origin have negative y-values.
Understanding the coordinate plane is crucial, as it provides a visual framework to plot points and graph linear equations effectively.
Linear Equations
Linear equations are mathematical expressions that describe a straight-line relationship between two variables. Each equation can usually be written in the form \(y = mx + b\), known as the slope-intercept form.
Here, \(m\) stands for the slope of the line, which indicates how steep the line is, and \(b\) is the y-intercept, the point where the line crosses the y-axis. Key features of linear equations include:
  • The slope \(m\) is calculated as the 'rise over run,' meaning how much the line goes up (or down) for a horizontal move.
  • A positive slope means the line inclines upwards, while a negative slope implies it slopes downwards.
  • The y-intercept \((0, b)\) shows where the line starts when \(x = 0\).
When you can express a linear equation in the slope-intercept form, it becomes easier to graph it on the coordinate plane and interpret its behavior.
Graphing Techniques
Graphing linear equations involves plotting points on the coordinate plane and drawing a line that best represents the equation. The slope-intercept approach is one of the simplest graphing techniques.
Here are steps to graph using this method:
  • Start by identifying the y-intercept \((0, b)\) from the equation. Plot this point on the y-axis.
  • Use the slope \(m\) to find another point by moving rightward for the x-intercept and accordingly up or down for the y-intercept.
  • Remember: the slope \(m = \frac{rise}{run}\) tells you how to navigate from one point to another directly.
  • Once you have at least two points plotted, draw a straight line through these points. Extend the line across the plane and add arrows to indicate it continuous infinitely.
Understanding these graphing techniques simplifies solving and visualizing linear equations through graphical representation, making it a valuable skill in algebra and beyond.

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

For each pair of functions \(f(x)\) and \(g(x)\), find a. \(f(g(x))\) b. \(g(f(x))\) and c. \(f(f(x))\) $$ f(x)=x^{8} ; \quad g(x)=2 x+5 $$

The world population (in millions) since the year 1700 is approximated by the exponential function \(P(x)=522(1.0053)^{x}\), where \(x\) is the number of years since 1700 (for \(0 \leq x \leq 200\) ). Using a calculator, estimate the world population in the year: $$ 1750 $$

Identify each function as a polynomial, a rational function, an exponential function, a piecewise linear function, or none of these. (Do not graph them; just identify their types.) $$ f(x)=\frac{1}{x+2} $$

For each pair of functions \(f(x)\) and \(g(x)\), find a. \(f(g(x))\) b. \(g(f(x))\) and c. \(f(f(x))\) $$ f(x)=x^{2}+1 ; g(x)=x-\sqrt{x} $$

Graph each function. \(f(x)=3^{x}\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

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.