Problem 22
Question
Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope \(-1 / 3\) and \(y\) -intercept \((0,-1)\)
Step-by-Step Solution
Verified Answer
The equation in standard form is \( x + 3y = -3 \).
1Step 1: Identify the Provided Information
We are given the slope of the line as \( m = -\frac{1}{3} \) and the \( y \)-intercept as the point \( (0, -1) \).
2Step 2: Write the Slope-Intercept Form
The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. Substituting the given values, we write:\[ y = -\frac{1}{3}x - 1 \]
3Step 3: Convert to Standard Form
The standard form of a linear equation is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers. To convert from slope-intercept form, first eliminate fractions by multiplying the entire equation by 3:\[ 3y = -x - 3 \]Reorganize the terms to solve for \( x \) and \( y \) on one side:\[ x + 3y = -3 \]
4Step 4: Verify the Standard Form
Ensure that the equation is in the standard form \( Ax + By = C \) and that \( A \), \( B \), and \( C \) are integers. Here, the equation \( x + 3y = -3 \) correctly follows this format with \( A = 1 \), \( B = 3 \), and \( C = -3 \).
Key Concepts
Slope-Intercept FormStandard FormY-Intercept
Slope-Intercept Form
Linear equations are often expressed in what is called the slope-intercept form. This form is very handy when you want to quickly identify the slope of a line and its point of intersection with the y-axis, known as the y-intercept. The slope-intercept form is given by:
The slope \( m \) describes how steep the line is. If \( m \) is positive, the line slopes upwards. If \( m \) is negative, as in our example with \( m = -\frac{1}{3} \), the line slopes downwards.
To form the equation of a line in slope-intercept form, substitute \( m \) and \( b \) into the formula. For the exercise, using the values \( m = -\frac{1}{3} \) and \( b = -1 \), the equation becomes:
\[ y = -\frac{1}{3}x - 1 \]Understanding this form makes it easier to graph linear equations and analyze their properties.
- \( y = mx + b \)
The slope \( m \) describes how steep the line is. If \( m \) is positive, the line slopes upwards. If \( m \) is negative, as in our example with \( m = -\frac{1}{3} \), the line slopes downwards.
To form the equation of a line in slope-intercept form, substitute \( m \) and \( b \) into the formula. For the exercise, using the values \( m = -\frac{1}{3} \) and \( b = -1 \), the equation becomes:
\[ y = -\frac{1}{3}x - 1 \]Understanding this form makes it easier to graph linear equations and analyze their properties.
Standard Form
After expressing a linear equation in slope-intercept form, converting it to standard form is often necessary for various applications. The standard form of a linear equation is written as:
To convert from the slope-intercept form to the standard form, follow these steps:
Then, reorganize to get the standard form: \( x + 3y = -3 \). With this process, the equation is ready for interpretation or further use in calculations.
- \( Ax + By = C \)
To convert from the slope-intercept form to the standard form, follow these steps:
- Eliminate any fractions by multiplying every term by the denominator.
- Rearrange the equation so that both \( x \) and \( y \) terms are on one side.
Then, reorganize to get the standard form: \( x + 3y = -3 \). With this process, the equation is ready for interpretation or further use in calculations.
Y-Intercept
The y-intercept is an essential concept in understanding linear equations. It indicates the point where the line crosses the y-axis. In mathematical terms, the y-intercept occurs when the x-coordinate is zero, denoted as \( (0, b) \).
In everyday terms, the y-intercept can be thought of as the starting point of a line on a graph. For example, if you have a line represented by \( y = mx + b \), the value \( b \) is the y-intercept.
In our exercise, the y-intercept is \( (0, -1) \), meaning that the line crosses the y-axis at -1. This information is straightforward and can be quickly read in the slope-intercept form equation: \( y = -\frac{1}{3}x - 1 \).
Understanding the y-intercept helps in sketching the graph of the line and in showing how one variable depends on another in real-world scenarios. For instance, knowing the y-intercept could help predict starting values or initial conditions in a problem situation.
In everyday terms, the y-intercept can be thought of as the starting point of a line on a graph. For example, if you have a line represented by \( y = mx + b \), the value \( b \) is the y-intercept.
In our exercise, the y-intercept is \( (0, -1) \), meaning that the line crosses the y-axis at -1. This information is straightforward and can be quickly read in the slope-intercept form equation: \( y = -\frac{1}{3}x - 1 \).
Understanding the y-intercept helps in sketching the graph of the line and in showing how one variable depends on another in real-world scenarios. For instance, knowing the y-intercept could help predict starting values or initial conditions in a problem situation.
Other exercises in this chapter
Problem 22
Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=-2 \cos (\pi x / 4) $$
View solution Problem 22
Use a graphing calculator to graph \(f(x)=x^{3}, x \geq 0\), and \(g(x)=x^{5}, x \geq 0\), together. When is \(f(x)>g(x)\), and when is \(f(x)
View solution Problem 23
Explain how the following functions can be obtained from \(y=x^{2}\) by basic transformations: (a) \(y=x^{2}-2\) (b) \(y=(x-1)^{2}+1\) (c) \(y=-2(x+2)^{2}\)
View solution Problem 23
Graph \(y=x^{n}, x \geq 0\), for \(n=1,2,3\), and 4 in one coordinate system. Where do the curves intersect?
View solution