Problem 24 - Elementary Algebra

Question

Graph the equations. $$ x=0 $$

Step-by-Step Solution

Answer

Answer: The equation x=0 represents a vertical line that passes through the origin. To graph it, plot the point (0,0) on a coordinate plane, draw a straight line extending from the intercept (0,0) upwards and downwards, and label the line with the given equation "x=0".

Step 1: Identify the type of line

The given equation is x=0, which represents a vertical line because the value of x remains constant (zero) and the value of y can be any real number.

Step 2: Find the intercepts

The x-intercept occurs when y = 0. For the given equation, x is always 0 so the x-intercept is (0,0). To find the y-intercept, set x = 0, which results in the equation already given. Thus, the y-intercept is also (0,0).

Step 3: Plot the intercepts

On a coordinate plane, plot the point (0,0), which is our x and y-intercept.

Step 4: Draw the line

Since the line is vertical and passes through the intercept, draw a straight line extending from the intercept (0,0) upwards and downwards. The line x=0 should pass through the origin and have an infinite length in both positive and negative y-directions.

Step 5: Label the line

Label the line with the given equation "x=0" to finish graphing the equation.

Key Concepts

Vertical Line

A vertical line is quite straightforward to identify on a coordinate plane. It is characterized by having a constant x-value, regardless of the y-value. In simpler terms, a vertical line goes straight up and down. For example, in the equation

• \(x = 0\),

x is fixed at 0, which means every point on this line has an x-coordinate of 0 while y can be any real number.
This is different from horizontal lines, where the y-value remains constant. Vertical lines are crucial for understanding and interpreting graphs since they help delineate boundaries horizontally for graphing operations or to show certain constraints within a graph.

Coordinate Plane

The coordinate plane is a two-dimensional tool used for graphing mathematical equations. It comprises two axes that intersect, typically at a right angle, displaying a grid format that helps in plotting points, lines, and curves. The two axes are:

• The x-axis (horizontal line)
• The y-axis (vertical line)

These axes divide the plane into four quadrants, making it easier to locate points. Each point on this plane is denoted by a pair of coordinates

• \((x, y)\)

where x indicates horizontal movement and y indicates vertical movement. Understanding the coordinate plane is fundamental because it allows you to visualize and solve graphical equations easily. When graphing a line, such as

• \(x = 0\),

you place points on this plane and draw connections to form shapes and lines.

Intercepts

Intercepts are vital for understanding where a line crosses the axes in a coordinate plane. The intercepts are denoted as:

• x-intercept: Where the line crosses the x-axis
• y-intercept: Where the line crosses the y-axis

For a vertical line like

• \(x = 0\),

the situation is unique. Both the x-intercept and y-intercept are at the origin,

• \((0, 0)\).

This is because the line goes directly through the intersection of the axes. To identify intercepts in other equations, you set either x or y to zero in the equation and solve for the remaining variable. Each intercept provides a critical anchor from which to begin graphing or analyzing a line.

Graphing Linear Equations

Graphing linear equations is the process of representing equations as straight lines within a coordinate plane. To graph a simple linear equation such as

• \(x = 0\),

follow these basic steps:
1. Identify the type of line. In this case, it's vertical, given by the equation.
2. Determine the intercepts. For vertical lines, the x and y intercept will often coincide at a single point, which is here at

• \((0, 0)\).

3. Plot the intercepts on the coordinate plane.
4. Draw a line through the intercept point extending infinitely in both the upward and downward directions.
5. Label the line with its corresponding equation.
Graphing linear equations helps in visualizing relationships and solving systems of equations, making it a vital skill in mathematics and many applied sciences.