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Chapter 5: Problem 45

Identify the slope and \(y\) -intercept, then graph the line. $$x+y=0$$

Short Answer

Expert verified
The slope of the given equation \(x+y=0\) is -1, and the y-intercept is 0. To graph the line, plot the y-intercept at the origin (0, 0) and use the slope to determine that the line will pass through points (1, -1) and (-1, 1). Connect these points and extend the line infinitely in both directions.

Step by step solution

01

Convert equation to slope-intercept form

To convert the equation x + y = 0 into slope-intercept form, we will isolate y: \[y = -x\] Now we have the equation in slope-intercept form, y = -x + 0.
02

Identify the slope and y-intercept

From the previous step, we have the equation in the form y = mx + b, where m represents the slope and b represents the y-intercept. In this case, we have m = -1 and b = 0. So, the slope is -1 and the y-intercept is 0.
03

Graph the line

To graph the line, follow these steps: 1. Begin with a Cartesian plane. 2. Mark the origin (0, 0) on the plane as it is the y-intercept. 3. Since the slope is -1, we know that the line makes a 45-degree angle with both the x and y-axes. Therefore, it will pass through the point (1, -1) and (-1, 1). 4. Connect the points on the graph, and use a straight edge to form the line extending infinitely in both directions. The line representing x + y = 0 has been graphed with a slope of -1 and y-intercept of 0.

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Key Concepts

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

graphing linear equations
Graphing linear equations is a fundamental skill in algebra that helps you visualize the relationship between two variables. A linear equation is an equation between two variables that gives a straight line when plotted on a graph. The standard form of a linear equation is the slope-intercept form, which looks like this: \( y = mx + b \).
Here, \( m \) stands for the slope of the line, and \( b \) represents the y-intercept.
Graphing involves several key steps:
  • Convert the given equation to slope-intercept form, if it isn't already.
  • Identify the slope (\( m \)) and the y-intercept (\( b \)).
  • Plot the y-intercept on the graph.
  • Use the slope to determine another point on the graph.
  • Draw a straight line through the points, extending it in both directions.
This visual representation makes it much easier to understand the behavior of the equation.
identifying slope and y-intercept
In order to graph a linear equation, it is crucial to identify the slope and y-intercept. The slope indicates how steep the line is and in which direction it travels.
The y-intercept is the point where the line crosses the y-axis. These two elements are found in the slope-intercept form: \( y = mx + b \).

To determine them:
  • Identify the coefficient of \( x \), which is the slope \( (m) \).
  • Find the constant term, which is the y-intercept \( (b) \).
In the case of the equation \( y = -x + 0 \), the slope \( (m) \) equals \(-1\), and the y-intercept \( (b) \) is \( 0 \).
This tells us that the line decreases one unit down the y-axis for every unit it moves right along the x-axis.
coordinate plane
A coordinate plane is a two-dimensional surface used to graph relationships between two variables.
It is made up of two perpendicular lines: the horizontal line is the x-axis, and the vertical line is the y-axis.
These axes divide the plane into four quadrants.

Here are some key aspects of the coordinate plane:
  • The point where the axes intersect is called the origin, represented as \((0, 0)\).
  • Each point on the plane has a pair of coordinates \((x, y)\).
  • The positive x-values are located to the right of the origin, and the negative to the left.
  • The positive y-values are above the origin, and the negative are below.
Graphing the line of an equation involves marking points that satisfy the equation on this plane, then connecting these points to create a visual representation of the line.

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Most popular questions from this chapter

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Write an equation of the line parallel to the given line and containing the given point. Write the answer in slopeintercept form or in standard form, as indicated. \(y=-3 x+1 ;(5,-19) ;\) slope-intercept form

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