/*! 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 45 Graph the line with the given eq... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 45

Graph the line with the given equation. \(y=2 x\)

Short Answer

Expert verified
The line passes through the origin and rises 2 units for each unit it moves to the right.

Step by step solution

01

Understanding the Equation

The equation given is in slope-intercept form: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For this equation, \(y = 2x\), the slope \(m = 2\) and the y-intercept \(b = 0\). This means the line passes through the origin (0,0) and has a slope of 2.
02

Plotting the Y-Intercept

Start by plotting the y-intercept on the graph. For the equation \(y = 2x\), the y-intercept is 0. So, place a point at the origin (0,0) on the graph.
03

Using the Slope to Find Another Point

The slope of the line is 2, which can be written as \(\frac{2}{1}\). This means for every 1 unit you move to the right on the x-axis, move 2 units up on the y-axis. From the origin, move 1 unit to the right to x = 1, then 2 units up to y = 2, and plot this point (1,2).
04

Drawing the Line

With the two points, (0,0) and (1,2), draw a straight line through both points, extending in both directions across the graph. This line represents the graph of the equation \(y = 2x\).
05

Checking Additional Points

To verify, choose another point by using the slope or plugging a value into the equation to see if it lies on the line. For example, when \(x = 2\), \(y = 2 \times 2 = 4\). Plot the point (2,4) to confirm it's on 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.

Slope-Intercept Form
Understanding the slope-intercept form is crucial for graphing linear equations. This form is written as \( y = mx + b \), where \( m \) denotes the slope, and \( b \) is the y-intercept.
In this formula, the variables \( x \) and \( y \) represent any point on the line, but it's \( m \) and \( b \) that give us the tools to graph efficiently.

  • The Slope (\( m \)): It tells us how steep the line is and the direction it goes, whether it's upwards or downwards.
  • The Y-Intercept (\( b \)): This is where the line crosses the y-axis. It's the starting point for graphing our line.
To graph using the slope-intercept form, you start at the y-intercept (\( b \)) and use the slope (\( m \)) to find other points on the line. This method simplifies plotting linear equations on a graph.
Y-Intercept
The y-intercept of a line is a vital component because it indicates where the line will touch the y-axis. For any linear equation in the slope-intercept form \( y = mx + b \), the y-intercept is the value of \( b \).
In simpler terms, it is the point where \( x \) is zero.

To visualize the concept:
  • Consider the point as "standing still" on the x-axis, as every change along x results in a different point on the line, except at the y-intercept.
  • For instance, in the equation \( y = 2x \), the value of \( b \) is zero. Therefore, the line starts at the origin point (0,0) on the graph because that鈥檚 where \( y \) equals zero.
Understanding the y-intercept helps set the starting point of any line you need to graph.
Slope
The slope is all about direction and steepness. In mathematics, it plays a significant role in understanding how lines behave on a graph.
For an equation, written in slope-intercept form \( y = mx + b \), the slope is represented by the coefficient \( m \).


What the slope tells us:
  • Rise over Run: The slope is calculated as \( \frac{{\text{rise}}}{{\text{run}}} \), meaning how many units the line goes up (rise) for a particular horizontal movement (run).
  • Direction: A positive slope means the line ascends as you move from left to right; a negative slope means it descends.
  • Steepness: A larger slope indicates a steeper line, while a smaller slope shows a flatter one.
For example, in the equation \( y = 2x \), the slope is \( 2 \). This means from any point on the line, moving 1 unit right on the x-axis, you move 2 units up on the y-axis. This rise over run helps pinpoint other points for accurate graphing.

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

Graph the line with the given equation. \(y=-\frac{1}{2} x\)

Solve each system of inequalities by graphing. $$ \begin{array}{l}{x+y<4} \\ {9 x^{2}-4 y^{2} \geq 36}\end{array} $$

Each equation is of the form \(A x^{2}+B x y+C y^{2}+D x+\) \(E y+F=0 .\) Identify the values of \(A, B,\) and \(C\). $$ x^{2}-4 x+4 y+2=0 $$

STRUCTURAL DESIGN An architect's design for a building includes some large pillars with cross sections in the shape of hyperbolas. The curves can be modeled by the equation \(\frac{x^{2}}{0.25}-\frac{y^{2}}{9}=1,\) where the units are in meters. If the pillars are 4 meters tall, find the width of the top of each pillar and the width of each pillar at the narrowest point in the middle. Round to the nearest centimeter.

What is an equation for the hyperbola centered at the origin with a vertical transverse axis of length 12 units and a conjugate axis of length 4 units?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

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.