/*! 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 73 Graph each equation \(y-3 x=-\... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 73

Graph each equation \(y-3 x=-\frac{4}{3}\)

Short Answer

Expert verified
Graph the line by plotting the y-intercept (-4/3) and using the slope 3 to find another point.

Step by step solution

01

Rewrite the equation in slope-intercept form

The slope-intercept form of a line's equation is given by \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. Start by adding \(3x\) to both sides of the given equation to isolate \(y\) on one side: \[y = 3x - \frac{4}{3}\].
02

Identify the slope and y-intercept

From the equation \(y = 3x - \frac{4}{3}\), identify the slope (\(m\)) and the y-intercept (\(b\)). Here:\[m = 3\] and \[b = -\frac{4}{3}\].
03

Plot the y-intercept on the graph

The y-intercept is the point where the line crosses the y-axis. In this case, the y-intercept is \(b = -\frac{4}{3}\). Plot the point \(0, -\frac{4}{3}\) on the graph.
04

Use the slope to find another point

The slope \(m = 3\) means that for every 1 unit increase in \(x\), \(y\) increases by 3 units. From the y-intercept at \(0, -\frac{4}{3}\), move 1 unit right to \(1, -\frac{4}{3}+3\) = \(1, \frac{5}{3}\). Plot this point on the graph.
05

Draw the line through the points

Now that you have the points \(0, -\frac{4}{3}\) and \(1, \frac{5}{3}\) plotted, draw a straight line through these points. This line represents the graph of the equation \(y = 3x - \frac{4}{3}\).

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Key Concepts

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

Understanding the Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most commonly used formats for expressing a straight line. It is given by the formula:
  • \( y = mx + b \)
where:
  • \( m \) is the slope of the line, indicating the steepness or direction of the line.
  • \( b \) is the y-intercept, showcasing where the line crosses the y-axis.
This format is highly convenient for graphing purposes because it immediately reveals the slope and the y-intercept, helping to quickly sketch the line. When graphing using the slope-intercept form, you first plot the y-intercept on the graph. This gives you a starting point for your line. Then, using the slope, you determine another point by moving according to the slope's rise over run, which allows you to draw the entire line confidently.
The Role of the Y-Intercept
The y-intercept in the context of a linear equation is an essential component. It refers to the precise point where the line crosses the y-axis. In the slope-intercept equation \( y = mx + b \), the value of \( b \) serves as the y-intercept. To find this point on a graph, set \( x = 0 \) and solve for \( y \). This point is your y-intercept:
  • For the example equation \( y = 3x - \frac{4}{3} \), the y-intercept is \( -\frac{4}{3} \).
Understanding the y-intercept is crucial as it provides a foundational point from which the rest of the line can be drawn. Once this point is plotted, you can subsequently use the slope to find additional points along the line.
Exploring Linear Functions
A linear function is depicted by a straight line on a graph and adheres to the equation form \( y = mx + b \). This kind of function is one of the most basic and vital in mathematics. Key features of a linear function include:
  • Constant Rate of Change: The slope \( m \) indicates that for every unit increase in \( x \), \( y \) changes consistently by \( m \) units.
  • Straight Line Graph: Since the relationship between variables is linear, the graph is always a straight line.
Consider the linear equation \( y = 3x - \frac{4}{3} \) from the example. This linear function's slope \( m = 3 \) indicates a steep line, making \( y \) increase rapidly as \( x \) grows. Recognizing these characteristics can assist you in identifying and working with linear functions accurately in various mathematical contexts.

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Most popular questions from this chapter

Explain how to graph a function by plotting points.

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