/*! 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 36 Sketch the graph of the line sat... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 36

Sketch the graph of the line satisfying the given conditions. Passing through \((3,2)\) with slope 2

Short Answer

Expert verified
The equation of the line is \( y = 2x - 4 \).

Step by step solution

01

- Identify the slope-intercept form

The slope-intercept form of a line is given by the equation: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. The given slope is 2.
02

- Use the point to find the y-intercept

Substitute the given point \((3, 2)\) and the slope \( m = 2 \) into the slope-intercept form equation to find \( b \).Starting with \[ y = mx + b \]Substitute \( y = 2 \), \( x = 3 \), and \( m = 2 \):\[ 2 = 2(3) + b \]Simplify:\[ 2 = 6 + b \]Solve for \( b \):\[ b = 2 - 6 \]\[ b = -4 \]Therefore, the y-intercept \( b \) is -4.
03

- Write the equation of the line

With the slope \( m = 2 \) and the y-intercept \( b = -4 \), the equation of the line is:\[ y = 2x - 4 \]
04

- Plot the line

To sketch the graph, start by plotting the y-intercept \( (0, -4) \).Next, use the slope to find another point. The slope \( 2 \) means a rise of 2 units for every run of 1 unit. From \( (0, -4) \), move up 2 units and right 1 unit to plot \( (1, -2) \).Finally, draw a straight line through these points.

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 of a linear equation is a fundamental concept in graphing lines. The formula is expressed as: \[ y = mx + b \] Where:
  • \(m\): This is the slope of the line. It indicates how steep the line is. The slope tells you how much \(y\) changes for a change in \(x\).
  • \(b\): This represents the y-intercept of the line. It is the point where the line crosses the y-axis.
Understanding the slope-intercept form helps simplify the process of plotting linear equations. For this exercise, the given slope \(m\) is \(2\). This means that for every unit increase in \(x\), \(y\) will increase by 2 units.

The slope-intercept form is particularly useful because:
  • It makes it easy to graph the line just by knowing the slope and y-intercept.
  • It allows for quick identification of key features of the graph.
finding y-intercept
Finding the y-intercept \(b\) is critical as it gives you one of the key points to start your graph. You can find \(b\) by substituting a known point on the line and the slope \(m\) into the slope-intercept form equation.

For the given problem, we substitute \((3, 2)\) and the slope \(m = 2\) using the equation below:

\[ y = mx + b \]

\[ 2 = 2(3) + b \]
\[ 2 = 6 + b \]
The next step is to isolate \(b\) by subtracting 6 from both sides:
\[ b = 2 - 6 \]
\[ b = -4 \]

Thus, the y-intercept \(b\) is -4. This is the point where the line crosses the y-axis, and it helps in plotting the graph accurately.
plotting points
Plotting points is the final step to graphing a linear equation. Start with the y-intercept you found earlier, which in this case is \( (0, -4) \). From here, use the slope to plot additional points.

For the slope \( 2 \), it indicates a rise of 2 units for every run of 1 unit. So, starting from \( (0, -4) \):
  • Move up 2 units
  • Move right 1 unit to land at the next point \( (1, -2) \)
Continue this process to find more points if needed. For example: Starting at \( (1, -2) \), move:
  • Up 2 units to get \( -2 + 2 = 0 \)
  • Right 1 unit to get \( 1 + 1 = 2 \)
So your point will be \( (2, 0) \). After plotting these points, draw a straight line through them. This line represents your linear equation \( y = 2x - 4 \).

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

Sketch the graph of \(d=3 t+4\) using the horizontal axis for \(t\) values and the vertical axis for \(d\) values.

Sketch the graph of the given equation. Label the intercepts. $$6(y-1)=3(x+2)$$

Simplify the given expression. $$x^{2}\left(x^{3}\right)\left(x^{2}\right)^{3}$$

Write an equation of the line satisfying the given conditions. Passing through \((-1,4)\) and \((2,-2)\)

An escalator in a mall carries customers a vertical distance of 22 feet while traveling a horizontal distance of 58 feet. Find the grade of the escalator.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

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.