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Chapter 3: Problem 19

In the following exercises, graph by plotting points. $$ y=\frac{4}{3} x-5 $$

Short Answer

Expert verified
Plot (0, -5), (3, -1), and (6, 3), then draw a line through these points.

Step by step solution

01

Understand the Equation

The equation given is in the slope-intercept form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. Here, \(m = \frac{4}{3}\) and \(b = -5\).
02

Plot the y-Intercept

Firstly, plot the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis, which is the value of \(b\). In this case, \(b = -5\), so plot the point (0, -5).
03

Use the Slope to Find Another Point

The slope \(\frac{4}{3}\) tells us that for every 3 units we move to the right (positive direction of x-axis), we move 4 units up (positive direction of y-axis). From the point (0, -5), move 3 units to the right to x = 3, and then 4 units up to y = -1. Plot this point (3, -1).
04

Plot More Points if Needed

If more points are required for accuracy, continue using the slope to find additional points. From (3, -1), move 3 units right to x = 6, and 4 units up to y = 3. Plot (6, 3).
05

Draw the Line

Once sufficient points are plotted, use a ruler to draw a straight line through the points. This line is the graph of the equation \(y = \frac{4}{3} x - 5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
In algebra, the slope-intercept form is a way to write the equation of a line. The general formula is: \(y = mx + b\) Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept. This form is useful because it directly shows both the slope and where the line crosses the y-axis.
In the given equation, \(y = \frac{4}{3}x - 5\), \(m = \frac{4}{3}\) and \(b = -5\). This tells us everything we need to start plotting the graph of this line.
Plotting Points
To graph a linear equation like \(y = \frac{4}{3}x - 5\), you can start by plotting some key points. These points typically come from using the slope and y-intercept.
  • First, plot the y-intercept, which is the point where the line crosses the y-axis.
  • Next, use the slope to find additional points. The slope \(\frac{4}{3}\) means you move 4 units up for every 3 units you move to the right.
  • Plot more points as needed for accuracy. This helps in drawing a precise line.
Once you've plotted several points, use a ruler to draw a straight line through them. This line is the graph of the given equation.
Y-Intercept
The y-intercept is a crucial part of the slope-intercept form equation \(y = mx + b\). It's the point where the line crosses the y-axis. To find it, simply look at the value of \(b\) in the equation. In our example, \(b = -5\).
This means the line crosses the y-axis at (0, -5).
Start by plotting this point on the graph. It provides a starting location from which you can use the slope to find additional points.
Slope
The slope \(m\) in the slope-intercept form \(y = mx + b\) shows how steep the line is. It describes the change in y for each unit change in x. A slope of \(\frac{4}{3}\) means for every 3 units you move right, you move 4 units up.
  • From your starting point (0, -5), move 3 units right, and 4 units up to reach (3, -1).
  • Plot this point. Repeat the process to find more points like (6, 3).
Using the slope consistently helps in plotting a precise line that correctly represents the equation.

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