Problem 81 For each linear equation, a. giv... [FREE SOLUTION]

Chapter 1: Problem 81

For each linear equation, a. give the slope \(m\) and \(y\) -intercept \(b\), if any, and b. graph the line. . \(2 x+3 y=6\)

Short Answer

Expert verified

Slope \(m\) is \(-\frac{2}{3}\), y-intercept \(b\) is 2.

Step by step solution

Convert to Slope-Intercept Form

To find the slope and y-intercept, we first need to convert the equation from standard form to slope-intercept form. The standard form is given as \(2x + 3y = 6\). We solve for \(y\) by isolating it on one side of the equation.\ Subtract \(2x\) from both sides to get:\[3y = -2x + 6\] Then, divide each term by 3 to solve for \(y\):\[y = -\frac{2}{3}x + 2\] The equation is now in slope-intercept form \(y = mx + b\).

Identify Slope and Y-Intercept

Now that the equation is in slope-intercept form, \(y = -\frac{2}{3}x + 2\), we can identify the slope \(m\) and the y-intercept \(b\).\ The slope \(m\) is \(-\frac{2}{3}\) and the y-intercept \(b\) is 2.

Graph the Line

To graph the line, start by plotting the y-intercept on the graph. The y-intercept \(b = 2\) means the line crosses the y-axis at the point \((0, 2)\).\ From the y-intercept, use the slope \(-\frac{2}{3}\). The slope indicates that for every 3 units you move to the right, the line moves 2 units down (since the slope is negative). \ Beginning from the point \((0, 2)\), move right 3 units to \((3, 2)\), and then go down 2 units to \((3, 0)\). Plot this point.\ Draw a line through these points to represent the line \(y = -\frac{2}{3}x + 2\).

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 is a way of expressing a linear equation so that it is easy to identify the slope and the y-intercept of the line. This form is written as:

\( y = mx + b \)

Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept. The slope \( m \) tells us how steep the line is, and the y-intercept \( b \) indicates where the line crosses the y-axis. This form is particularly useful because it allows us to quickly visualize important characteristics of the line without having to do complex calculations.
In the exercise above, we transformed the equation from standard form to slope-intercept form to easily graph and analyze the characteristics of the line.

Standard Form

Standard form of a linear equation provides another way to express a line on a graph. It is generally written as:

\( Ax + By = C \)

where \( A \), \( B \), and \( C \) are integers, and \( A \) should be non-negative. This form can be advantageous when working with graphing or solving systems of linear equations, as it clearly shows the coefficients for both \( x \) and \( y \).
In the given exercise, the equation \( 2x + 3y = 6 \) is in standard form. To find the slope and y-intercept, it鈥檚 helpful to convert it to the slope-intercept form so calculations and graphing can be handled more easily.

Graphing Linear Equations

Graphing linear equations involves visually representing them on a coordinate plane. This is made simpler by starting with the slope-intercept form:

Identify the y-intercept \( b \) and plot it on the y-axis.
Use the slope \( m \) to determine other points on the line. A positive slope means the line goes up as it moves to the right, and a negative slope means it goes down.

In the exercise, after converting to the slope-intercept form \( y = -\frac{2}{3}x + 2 \), we identify that the line crosses the y-axis at the point \((0, 2)\). Then, using the slope of \(-\frac{2}{3}\), we move 3 units to the right and 2 units down to plot additional points and draw the line.

Slope and Y-Intercept

The slope and y-intercept are two of the most fundamental characteristics of a linear equation.

The slope \( m \) indicates the direction and steepness of the line. It's calculated as the rise over run, meaning the vertical change \( \Delta y \) over the horizontal change \( \Delta x \).
The y-intercept \( b \) is the point where the line hits the y-axis. It indicates the value of \( y \) when \( x = 0 \).

In the example exercise, the slope is \(-\frac{2}{3}\), which tells us the line falls as it moves from left to right. The y-intercept is at \((0, 2)\), making it easy to start plotting the line on a graph. Recognizing these parameters enables us to quickly understand and draw the graph of the linear equation.

91影视

For each linear equation, a. give the slope \(m\) and \(y\) -intercept \(b\), if any, and b. graph the line. . \(2 x+3 y=6\)

Short Answer

Step by step solution

Convert to Slope-Intercept Form

Identify Slope and Y-Intercept

Graph the Line

Key Concepts

Slope-Intercept Form

Standard Form

Graphing Linear Equations

Slope and Y-Intercept

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Geometry

Logic and Functions

Mechanics Maths

Pure Maths

Study anywhere. Anytime. Across all devices.