/*! 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 42 Graph the line that passes throu... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 42

Graph the line that passes through the given point and has the given slope. (Objective 3 ) $$(-1,0), m=\frac{3}{4}$$

Short Answer

Expert verified
Graph the line using the slope-intercept form: start at \((-1,0)\), then move up 3 and right 4 to plot another point and draw the line.

Step by step solution

01

Identify the point-slope form

To graph the line, we use the point-slope form of a linear equation, which is given by \( y - y_1 = m(x - x_1) \). Here, \( m = \frac{3}{4} \), \( x_1 = -1 \), and \( y_1 = 0 \).
02

Substitute values into the formula

Substitute the known values into the point-slope equation: \( y - 0 = \frac{3}{4}(x - (-1)) \), simplifying it to \( y = \frac{3}{4}(x + 1) \).
03

Simplify the equation

Expand the equation to make it easier to graph. Distribute \( \frac{3}{4} \) through the parenthesis: \( y = \frac{3}{4}x + \frac{3}{4} \).
04

Find the y-intercept

In the equation \( y = \frac{3}{4}x + \frac{3}{4} \), identify the y-intercept, which is the constant term \( \frac{3}{4} \). This means the line crosses the y-axis at \( y = \frac{3}{4} \).
05

Plot the point and draw the slope

Start by plotting the given point \((-1, 0)\) on the coordinate plane. Use the slope \( m = \frac{3}{4} \) to determine another point. From \((-1, 0)\), move 3 units up and 4 units to the right, landing at \((3, 3)\). Plot this second point.
06

Draw the line

Connect the points \((-1, 0)\) and \((3, 3)\) with a straight line. This line represents the equation \( y = \frac{3}{4}x + \frac{3}{4} \), with the given slope and passing through the specified point.

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.

Point-Slope Form
The point-slope form is a way to write the equation of a line. It is useful when you know the slope of a line and one point on the line. The form is written as:
  • \( y - y_1 = m(x - x_1) \)
Here, \(m\) represents the slope of the line, and \((x_1, y_1)\) is a given point on the line.
To use this form
  • Replace \(x_1\) and \(y_1\) with the coordinates of the point.
  • Replace \(m\) with the slope.
This gives you the equation specific to the line passing through the known point.
In our exercise, the point given is \((-1,0)\) and the slope is \(\frac{3}{4}\). So, we substitute these values into the formula like so:
  • \(y - 0 = \frac{3}{4}(x - (-1))\)
This form can then be simplified to help plot or understand the line's behavior.
Y-intercept
The y-intercept is where the line crosses the y-axis on a graph. It is an important feature of a linear equation written in slope-intercept form \(y = mx + b\), where \(b\) is the y-intercept.
In a graph, the y-intercept tells us at what point the line meets the vertical axis. It provides a starting point to draw the rest of the line when you have the slope.
In the equation from our exercise \(y = \frac{3}{4}x + \frac{3}{4}\):
  • The term \(\frac{3}{4}\) without the \(x\) is the y-intercept.
This means the line will cross the y-axis at \(y = \frac{3}{4}\). Knowing the y-intercept helps in quickly plotting the line on a graph because you immediately know a point on that line.
Slope
The slope of a line describes its steepness and direction, calculated as "rise over run." It is represented by \(m\) in the equation of a line. Slope is a crucial aspect because it tells you how the line rises or falls as you move along it.
  • A positive slope means the line rises from left to right.
  • A negative slope means the line falls from left to right.
  • A zero slope means the line is horizontal.
  • An undefined slope means the line is vertical.
The slope formula is \(m = \frac{\text{change in } y}{\text{change in } x}\).
For our exercise, the slope \(m\) is \(\frac{3}{4}\). This indicates that for every 4 units you move right along the x-axis, the line moves up 3 units. Using these small movements, you can plot the line by starting from one point and using the slope to find another point.

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

Find the equation of the line that contains the two given points. Express equations in the form \(A x+B y=C\), where \(A, B\), and \(C\) are integers. (Objective \(1 \mathrm{~b}\) ) \((-1,-2)\) and \((-6,-7)\)

Explain the importance of the slope-intercept form \((y=m x+b)\) of the equation of a line.

Find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. (Objective 1a) \(m=\frac{5}{9}\) and \(b=4\)

For Problems \(33-44\), determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. $$ 3 x-5 y=15 $$

Contains the point \((-2,-3)\) and is perpendicular to the line \(x+4 y=6\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

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.