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Chapter 3: Problem 69

Graph each equation. $$ y-3 x=-\frac{4}{3} $$

Short Answer

Expert verified
Graph the line with y-intercept at \(-\frac{4}{3}\) and slope of 3.

Step by step solution

01

Rewrite in Slope-Intercept Form

We begin by rewriting the given equation in the slope-intercept form, which is typically written as \( y = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept. Starting with the equation \( y - 3x = -\frac{4}{3} \), we can add \( 3x \) to both sides to get \( y = 3x - \frac{4}{3} \).
02

Identify the Slope and Y-Intercept

From the slope-intercept form \( y = 3x - \frac{4}{3} \), we can see that the slope \( m \) is 3 and the y-intercept \( b \) is \(-\frac{4}{3}\). This tells us that the line rises 3 units vertically for every 1 unit it moves horizontally, and it crosses the y-axis at \(-\frac{4}{3}\).
03

Plot the Y-Intercept on the Graph

To start graphing, plot the y-intercept of the line on the Cartesian plane. Place a point at \( (0, -\frac{4}{3}) \) on the y-axis.
04

Use the Slope to Plot a Second Point

Use the slope to find another point on the line. From \( (0, -\frac{4}{3}) \), move 3 units up and 1 unit to the right to reach the point \( (1, \frac{5}{3}) \). Plot this second point on the graph.
05

Draw the Line

Finally, draw a straight line passing through both points \( (0, -\frac{4}{3}) \) and \( (1, \frac{5}{3}) \). Extend this line in both directions to complete the graph of the equation.

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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 linear equations.Using the formula, it is written as \( y = mx + b \), where \( m \) and \( b \) have special meanings:
  • \( m \) is the slope of the line. This value tells us how steep the line is and in which direction it slants.
  • \( b \) is the y-intercept of the line. This is where the line crosses the y-axis.
Using the slope-intercept form makes it easy to graph the equation as both the slope and the y-intercept are immediately apparent.For instance, when we have the equation \( y - 3x = -\frac{4}{3} \), our first step is to translate it into slope-intercept form. Rearranging it gives us \( y = 3x - \frac{4}{3} \). This equation is now ready for graphing because we can clearly see the slope and the y-intercept.
Plotting Points
Once you have an equation in the slope-intercept form, plotting points becomes a straightforward task.The graph will start at the y-intercept and then use the slope to determine the location of the next point.To illustrate:
  • Start by plotting the y-intercept point of the graph on the Cartesian plane. This point is \((0, b)\), where the y-coordinate is the y-intercept value.
  • Next, use the slope to find subsequent points. The slope is a fraction representing the rise over the run. This tells you how many units to move vertically (up or down) and horizontally (left or right).
In the example \( y = 3x - \frac{4}{3} \):
  • Begin the graph by plotting the point \((0, -\frac{4}{3})\), which is our y-intercept.
  • The slope is 3, or \(3/1\), so from the y-intercept, move 3 units up and 1 unit to the right to find the second point, \((1, \frac{5}{3})\).
After plotting these two points, draw a line through them to represent the complete graph.
Slope
The slope of a line is a measure of its steepness.In the equation form \( y = mx + b \), \( m \) represents the slope.The slope describes the vertical change divided by the horizontal change between two points on the line.Mathematically it is expressed as \( \text{slope} = \frac{\text{rise}}{\text{run}} \).For a clearer visual:
  • If the slope \( m \) is positive, the line ascends from left to right. If it鈥檚 negative, the line descends.
  • In our equation \( y = 3x - \frac{4}{3} \), the slope \( m \) is 3. This means for every 1 unit the line moves horizontally, it moves up 3 units vertically.
Understanding slope is crucial for predicting how a change in one variable affects the other in the equation.It helps to identify patterns and relationships in the data represented graphically.
Y-Intercept
The y-intercept is where the graph of an equation intersects the y-axis.In the slope-intercept form \( y = mx + b \), the \( b \) value represents the y-intercept.This point is crucial as it provides a starting point on the graph.It is always located where the x-coordinate equals zero.Here鈥檚 why it matters:
  • The y-intercept is essentially the value of \( y \) when \( x = 0 \). It's at this point where the line crosses the vertical axis on a graph.
  • For the equation \( y = 3x - \frac{4}{3} \), the y-intercept is \(-\frac{4}{3}\). Thus, the line crosses the y-axis at this point.
Graphing always begins at the y-intercept, setting the foundation to apply the slope and extend the line accurately across the plane.

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Most popular questions from this chapter

Determine whether the lines through each pair of points are parallel, perpendicular, or neither. See Example 8. \((-1,8)\) and \((-6,8)\) \((3,3)\) and \((3,7)\)

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