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Chapter 3: Problem 67

Graph each equation. $$ x=-\frac{5}{3} $$

Short Answer

Expert verified
The graph is a vertical line crossing the x-axis at \( x = -\frac{5}{3} \).

Step by step solution

01

Identify the Equation Type

The equation given is \( x = -\frac{5}{3} \), which specifies the value of \( x \) as a constant. This indicates that it represents a vertical line on the coordinate plane.
02

Understand Vertical Lines

Vertical lines have the equation form \( x = a \), where \( a \) is a constant. In this scenario, \( x = -\frac{5}{3} \) is a vertical line that intersects the x-axis at \( x = -\frac{5}{3} \).
03

Graph the Vertical Line

To graph \( x = -\frac{5}{3} \), draw a straight vertical line through the point on the x-axis where \( x = -\frac{5}{3} \). Since it's a vertical line, it will extend infinitely up and down, parallel to the y-axis, but it will only cross the x-axis at the specified point.

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Key Concepts

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

Understanding the Coordinate Plane
The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves to represent mathematical equations visually. This plane consists of two perpendicular lines known as the axes: the horizontal line is the x-axis, and the vertical line is the y-axis. These axes intersect at the origin, which has the coordinates \(0, 0\).

Each point on the plane is described by a pair of numbers \(x, y\), where \(x\) represents the horizontal position, and \(y\) represents the vertical position.
  • The coordinate plane is divided into four quadrants, each determined by the signs of \(x\) and \(y\).
  • Quadrants are counted counter-clockwise starting from the top right.
Understanding how to navigate this plane is crucial for graphing all types of lines and curves, including vertical lines like \(x = -\frac{5}{3}\).
Characteristics of a Vertical Line
A vertical line on the coordinate plane is represented by the equation \(x = a\), where \(a\) is a constant value. Unlike other lines, a vertical line does not have a slope.

Here are the key features of a vertical line:
  • It runs straight up and down, parallel to the y-axis.
  • It has no change in \(y\) as it moves horizontally, which is why the slope is considered undefined.
  • It remains at a constant \(x\) value, meaning every point on the line has the same \(x\) coordinate.
To graph \(x = -\frac{5}{3}\), you simply draw a line that passes through every point where the x-coordinate is \(-\frac{5}{3}\), spanning infinitely in both the upward and downward directions.
Intersection with the X-Axis
The intersection of a line with the x-axis is where the line crosses or touches the x-axis. This point is significant because it provides valuable information about the line's equation.

For a vertical line like \(x = -\frac{5}{3}\), its interaction with the x-axis is unique:
  • It intersects the x-axis at exactly one point, which is \((-\frac{5}{3}, 0)\).
  • This means on the x-axis, the line touches precisely where \(x\) is \(-\frac{5}{3}\).
  • Since there is no \(y\) value involved in the equation for a vertical line, it won't extend horizontally along the x-axis.
This intersection point is crucial when graphing, as it provides a visual cue to position the line on the plot accurately.

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Most popular questions from this chapter

Find the slope and the y-intercept of the graph of each equation and graph it. See Examples 4 and 5. $$ y=\frac{x}{3} $$

Find the slope of a line perpendicular to the line passing through the given two points. See Example \(9 .\) \((0,0)\) and \((5,-9)\)

Halloween Candy. A candy maker wants to make a 60 -pound mixture of two candies to sell for $$ 2\( per pound. If black licorice bits sell for \) 1.90 per pound and orange gumdrops sell for $$ 2.20$ per pound, how many pounds of each should be used?

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

How do we distinguish between a line with positive slope and a line with negative slope?
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