Problem 49 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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 \(x=-3\).

Key Concepts

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’t 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.