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Chapter 2: Problem 49

Graph each equation in the rectangular coordinate system. $$x=-3$$

Short Answer

Expert verified
The graph of the equation \(x=-3\) is a vertical line passing through the point \(-3\) on the x-axis.

Step by step solution

01

Identify the type of line

First recognize the equation format. Since there is no y-term in the provided equation, this is a vertical line.
02

Identify the intercept

Since our equation is \(x=-3\), the x-intercept (where the line intersects the x-axis) is -3. Note that vertical lines have an x-intercept, but no y-intercept.
03

Draw the line

Draw a vertical line on the graph passing through the point \(-3\) on the x-axis. This line represents all points whose x-coordinate is \(-3\), aligning with the equation \(x=-3\).

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Key Concepts

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

Vertical Lines
Vertical lines are a fundamental concept when graphing linear equations. They are unique because they extend straight up and down along the y-axis.
Vertical lines are defined by equations that have the form \(x = c\), where \(c\) is a constant. For instance, the equation \(x = -3\) describes a vertical line that intersects the x-axis at -3.
  • Vertical lines are characterized by their constant x-values. This means that every point on a vertical line has the same x-coordinate.
  • Such lines do not have a slope, or you could say they have an "undefined" slope.
  • Vertical lines do not intersect the y-axis, so they don't have a y-intercept. This property distinguishes them from most other types of lines in a graph.
Understanding vertical lines is vital to identifying graphs correctly and interpreting equations efficiently.
Rectangular Coordinate System
The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional system used for graphing equations.
It consists of two perpendicular number lines, the x-axis and the y-axis, which intersect at the origin \( (0,0) \).
  • The x-axis runs horizontally and typically represents the independent variable in a graph.
  • The y-axis runs vertically and usually corresponds to the dependent variable.
  • Every point in this system is defined by an ordered pair \((x, y)\), which specifies its horizontal and vertical positions.
This system simplifies the process of plotting equations by providing a clear framework. For example, with the equation \(x = -3\), the rectangular coordinate system helps visualize this as a vertical line intersecting the x-axis at -3.
X-intercept
The x-intercept is the point where a graph intersects the x-axis. For linear equations, this is a crucial concept as it helps to easily identify where the line crosses or touches the x-axis on a graph.
In simpler terms, it's the value of \(x\) when \(y\) is 0.
  • For vertical lines, such as \(x = -3\), the entire line represents the x-intercept since it runs parallel to the y-axis and doesn鈥檛 extend across it.
  • Regular linear equations will often have a specific single x-intercept, but with vertical lines, every point on the line could be considered an x-intercept.
  • This makes vertical lines particularly special as their intercept is not just a single point but a line itself lying on the x-coordinate provided by the equation.
Mastering the concept of x-intercepts allows for a deeper understanding of how linear equations interact with the axes in a coordinate system.

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