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Chapter 3: Problem 60

Graph each equation. \(x+1=0 \quad\) (Hint: Solve for \(x\) first.)

Short Answer

Expert verified
The graph of the equation is a vertical line at \(x = -1\).

Step by step solution

01

Isolate the Variable

Start by rearranging the equation to solve for the variable. The given equation is \(x + 1 = 0\). Subtract 1 from each side to isolate \(x\): \(x = -1\).
02

Identify the Type of Equation

Recognize that \(x = -1\) is a linear equation representing a vertical line on the Cartesian plane. The equation signifies that \(x\) is constant for all values of \(y\).
03

Plot the Graph

On a Cartesian plane, the equation \(x = -1\) is represented as a vertical line. To plot this, find the line where \(x\) is -1. The line is parallel to the y-axis and vertical through the point -1 on the x-axis. All points on this line have x-coordinates of -1 (e.g., (-1, 0), (-1, 1), (-1, -1)).

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Key Concepts

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

Cartesian Plane
The Cartesian plane is a two-dimensional grid system that helps visualize mathematical concepts like equations. It's made up of two intersecting lines called the axes: the horizontal line (x-axis) and the vertical line (y-axis).
  • The point where these axes intersect is called the origin, labeled as (0, 0).
  • Each point on the plane is defined by a pair of numbers written as \((x, y)\).
  • The x-value shows the position along the horizontal axis, and the y-value shows the position along the vertical axis.
Think of the Cartesian plane as a map that helps us pinpoint exact locations using these x and y coordinates. This makes it easier to understand relationships and patterns in math, especially when graphing linear equations.
Vertical Line
A vertical line on the Cartesian plane has some unique characteristics. It stands upright and runs parallel to the y-axis. Here are some key points:
  • A vertical line represents a constant x-value, meaning no matter where you look on the line, the x-coordinate is always the same.
  • For instance, when graphing the equation \(x = -1\), it creates a vertical line where every point on this line has the x-coordinate of -1, while the y-coordinate can vary.
  • Vertical lines are especially interesting because they do not have a slope. This makes them distinct from most other lines which incline up or down.
Understanding vertical lines helps in graphing linear equations and recognizing their properties, such as being undefined in terms of slope.
Linear Equation
A linear equation is an equation that plots as a straight line on the Cartesian plane. These equations usually come in the form \(y = mx + c\), where:
  • \(m\) is the slope, indicating the tilt or incline of the line.
  • \(c\) is the y-intercept, telling us where the line crosses the y-axis.
However, not all linear equations follow this format, especially ones like \(x = -1\), which represent vertical lines. This equation is still linear but differs because:
  • It does not have a y-variable, instead, it shows that the x-value is constant.
  • Lines like these are vertical, representing an undefined slope, unlike most linear equations where slopes are defined.
Grasping linear equations is fundamental in graphing and solving mathematical problems, as they form the backbone of many real-world scenarios.

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Most popular questions from this chapter

The length of a rectangular picture is 5 inches greater than twice the width. If the perimeter is 112 inches, find the dimensions of the frame.

Sewing costs. A tailor charges a basic fee of \(\$ 20\) plus \(\$ 5\) per letter to sew an athlete's name on the back of a jacket. Write a linear equation that will find the cost to have a name containing \(x\) letters sewn on the back of a jacket.

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Find the slope and the y-intercept of the graph of each equation and graph it. See Examples 4 and 5. $$ y=-4 x $$

Find the slope of each line. See Examples 4 and \(5 .\) $$3 x=-12$$
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