/*! 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 For each line, ( \(a\) ) find th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 45

For each line, ( \(a\) ) find the slope and ( \(b\) ) sketch the graph. $$y=3 x+5$$

Short Answer

Expert verified
The slope is 3, and the graph is a straight line passing through the points (0, 5) and (1, 8).

Step by step solution

01

Identify the Equation Type

The given equation is in the slope-intercept form, which is written as \[ y = mx + b \]. In this equation, \( m \) represents the slope and \( b \) represents the y-intercept.
02

Determine the Slope

Compare the given equation to the slope-intercept form. Here, we have \[ y = 3x + 5 \]. The coefficient of \( x \) is the slope. Therefore, the slope \( m \) is 3.
03

Find the Y-Intercept

The constant term in the equation is the y-intercept \( b \). Therefore, the y-intercept is 5.
04

Plot the Y-Intercept

Start by plotting the y-intercept on the graph. The point (0, 5) is located directly on the y-axis.
05

Use the Slope to Find Another Point

The slope \( m \) is the ratio of the rise over the run. For a slope of 3, it can be written as \[ m = \frac{3}{1} \], which means for every 1 unit moved to the right, move up 3 units. From the y-intercept (0, 5), moving right 1 unit takes you to (1, 5), and moving up 3 units takes you to the point (1, 8).
06

Draw the Line

Draw a straight line through the points (0, 5) and (1, 8). This is the graph of the equation \( y = 3x + 5 \).

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.

line graph
A line graph is a visual representation of a linear equation on a coordinate system. It helps to understand the relationship between variables easily. When plotting a line graph:
- Start with a coordinate system with x and y axes.
- Identify important points from the equation, like the y-intercept.
- Use these points to plot the graph accurately.
A good basis starts with recognizing key characteristics of the linear equation in slope-intercept form, which we will cover next.
slope calculation
To calculate the slope of a line, we use the coefficient of x in the slope-intercept form equation given as \(y = mx + b\). Here, the slope (m) tells us how steep the line is.
- For the equation \(y = 3x + 5\), compare it to \(y = mx + b\).
- The coefficient of x (3) is the slope.

Let's understand slope more deeply:
  • Positive Slope: Indicates the line rises as you move from left to right.
  • Negative Slope: Indicates the line falls as you move from left to right.
  • Zero Slope: Produces a horizontal line.
  • Undefined Slope: Produces a vertical line.
The slope of 3 means 'rise 3 units for every 1 unit run'. This ratio helps plot additional points accurately.
y-intercept
The y-intercept of a line is where the graph crosses the y-axis. It is represented by \(b\) in the slope-intercept form equation \(y = mx + b\). For example:
- In the equation \(y=3x+5\), the constant term 5 is the y-intercept.
- This means the line crosses the y-axis at point (0, 5).

Key steps to using the y-intercept in graphing:
  • Plot the y-intercept on the coordinate plane.
  • Use the slope to find other points starting from the y-intercept.
For the equation \(y = 3x + 5\), start at (0, 5), then use the slope (3) to determine the next points. Move 1 unit right and 3 units up to get to (1, 8). Connect these points to draw 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

Given functions \(f\) and \(g,\) find ( \(a\) ) \((f \circ g)(x)\) and its domain, and ( \(b\) ) \((g \circ f)(x)\) and its domain. See Examples 6 and 7 . $$f(x)=\sqrt{x}, \quad g(x)=x+3$$

Given functions \(f\) and \(g,\) find ( \(a\) ) \((f \circ g)(x)\) and its domain, and ( \(b\) ) \((g \circ f)(x)\) and its domain. See Examples 6 and 7 . $$f(x)=\sqrt{x}, \quad g(x)=\frac{3}{x+6}$$

Find the center-radius form of the circle described or graphed. a circle having a diameter with endpoints \((-1,2)\) and \((11,7)\)

For each of the functions in Exercises \(33-46,\) find ( \(a\) ) \(f(x+h),\) (b) \(f(x+h)-f(x)\) and \((c) \frac{f(x+h)-f(x)}{h} .\) See Example 4. $$f(x)=6 x+2$$

Use a graphing calculator to solve each linear equation. $$3(2 x+1)-2(x-2)=5$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

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.