Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 7: Problem 20
Graph each equation using the slope and \(y\) -intercept. $$x+y=0$$
Short Answer
Expert verified
Graph the line through (0, 0) and (1, -1) on a coordinate plane.
Step by step solution
01
Write the Equation in Slope-Intercept Form
The slope-intercept form of an equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. Start by solving for \( y \) in the equation \( x + y = 0 \). Subtract \( x \) from both sides to get \( y = -x \). This equation is already in slope-intercept form with \( m = -1 \) and \( b = 0 \).
02
Identify the Slope and Y-Intercept
From the equation \( y = -x \), we see that the slope \( m \) is \(-1\), and the \( y \)-intercept \( b \) is \(0\). The line crosses the \( y \)-axis at the point \( (0, 0) \).
03
Graph the Y-Intercept
Plot the \( y \)-intercept on the graph. This is the point where the line crosses the \( y \)-axis. In this case, plot the point \( (0, 0) \).
04
Use the Slope to Plot Another Point
The slope \(-1\) can be interpreted as \(-1/1\). From the \( y \)-intercept at \( (0, 0) \), move down 1 unit and right 1 unit to locate the next point, which is \( (1, -1) \). Plot this point on the graph.
05
Draw the Line
With the points \( (0, 0) \) and \( (1, -1) \) plotted, draw a straight line through these points. This line represents the equation \( x + y = 0 \). Ensure the line extends in both directions and passes through both points.
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
Graphing linear equations can be made easier by using the slope-intercept form. This form of a linear equation 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 \( b \) is the point where the line crosses the \( y \)-axis. To express an equation in this form, sometimes you need to rearrange it to solve for \( y \).
In our exercise, the equation \( x + y = 0 \) was transformed into slope-intercept form by subtracting \( x \) from both sides, resulting in \( y = -x \). Once in this shape, it's easy to spot the slope and \( y \)-intercept. In this example, the slope is \(-1\) and the \( y \)-intercept is \(0\). This reformulation of equations is the first critical step in graphically representing a line.
In our exercise, the equation \( x + y = 0 \) was transformed into slope-intercept form by subtracting \( x \) from both sides, resulting in \( y = -x \). Once in this shape, it's easy to spot the slope and \( y \)-intercept. In this example, the slope is \(-1\) and the \( y \)-intercept is \(0\). This reformulation of equations is the first critical step in graphically representing a line.
Y-Intercept
The \( y \)-intercept is an essential part of understanding and graphing linear equations. It is the point where the line crosses the \( y \)-axis on a graph. In mathematical terms, it's the value of \( y \) when \( x = 0 \).
In the context of the specific equation \( y = -x \), the \( y \)-intercept is \( (0, 0) \). This means the line passes through the origin. Locating the \( y \)-intercept helps to anchor the graph. By plotting this point first, you create a start for your line, making it easier to choose additional points based on the slope for further accuracy.
To locate the \( y \)-intercept in any equation, simply substitute \( x = 0 \) into the expression and solve for \( y \). Then, mark this point on the graph. This helps ensure the line accurately represents the solution of the equation.
In the context of the specific equation \( y = -x \), the \( y \)-intercept is \( (0, 0) \). This means the line passes through the origin. Locating the \( y \)-intercept helps to anchor the graph. By plotting this point first, you create a start for your line, making it easier to choose additional points based on the slope for further accuracy.
To locate the \( y \)-intercept in any equation, simply substitute \( x = 0 \) into the expression and solve for \( y \). Then, mark this point on the graph. This helps ensure the line accurately represents the solution of the equation.
Plotting Points
Once you've found the \( y \)-intercept, the next step is plotting additional points using the slope. The slope, denoted as \( m \), is a fraction \( \Delta y / \Delta x \), where \( \Delta y \) is the change in \( y \), and \( \Delta x \) is the change in \( x \).
For the equation \( y = -x \), the slope is \(-1\), which can be rewritten as \(-1/1\). Starting from the \( y \)-intercept \( (0, 0) \), move down 1 unit in the \( y \) direction and right 1 unit in the \( x \) direction to find the next point at \( (1, -1) \). Plot this point on your graph as well.
By connecting these points with a straight line, you visually demonstrate the relation between \( x \) and \( y \) in your equation. Drawing multiple unique points using slope helps ensure accuracy in your graph and confirms that the plotted line is correct.
For the equation \( y = -x \), the slope is \(-1\), which can be rewritten as \(-1/1\). Starting from the \( y \)-intercept \( (0, 0) \), move down 1 unit in the \( y \) direction and right 1 unit in the \( x \) direction to find the next point at \( (1, -1) \). Plot this point on your graph as well.
By connecting these points with a straight line, you visually demonstrate the relation between \( x \) and \( y \) in your equation. Drawing multiple unique points using slope helps ensure accuracy in your graph and confirms that the plotted line is correct.
