Problem 57 - Algebra 1: Concepts and Skills

Question

Determine whether the line is horizontal or vertical. Then graph the line. $$x=4$$

Step-by-Step Solution

Answer

The line \(x=4\) is a vertical line. In the graph, it is presented as a straight line located at 4 on the x-axis.

Step 1: Identify the type of line

The equation given is \(x=4\). In an \(x=\)constant equation, the 'x' coordinate is the same for any 'y' coordinate. This indicates that it is a vertical line because all points on the line have the same 'x' value.

Step 2: Draw the line

To graph a vertical line: First, identify the value of \(x\), which we know is 4. On a graph, locate the number 4 on the x-axis. Then draw a straight vertical line through this point. This line represents all points where \(x=4\), which is exactly our given equation.

Key Concepts

Vertical Line

A vertical line is a straight line that runs up and down, parallel to the y-axis. It is characterized by having a constant x-coordinate for all points on the line. In the equation form, vertical lines are represented as \(x = \text{constant}\). For example, in the equation \(x = 4\), every point on this line will have an x-value of 4, regardless of the y-value.

• Vertical lines do not have a slope because their direction is perfectly vertical.
• This means that they are perpendicular to the x-axis.

In coordinate geometry, vertical lines play a critical role as they help in understanding the behavior of points and shapes in a plane. Whether you're graphing simple equations or dealing with complex geometrical problems, recognizing vertical lines can simplify your task considerably.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where points are defined and their relationships are explored using a coordinate system. The most common coordinate system is the Cartesian coordinate system, which comprises two perpendicular axes: the x-axis and the y-axis.

• Points are expressed as pairs \((x, y)\), where 'x' is the horizontal value, and 'y' is the vertical value.
• Lines, such as our vertical line example \(x = 4\), help define relationships between these points.

By using equations to express these lines, coordinate geometry allows for precise calculations and visualizations. It is essential for graphing linear equations and is the foundation of more advanced mathematical topics. Understanding how to plot points and interpret lines on a graph is fundamental in mastering this subject.

X-axis

The x-axis is the horizontal line in the Cartesian coordinate plane. It runs left to right and intersects with the y-axis at the origin, labeled as point \((0, 0)\).

When graphing, the x-axis is the reference that determines the horizontal coordinates of points. It is crucial when plotting lines, especially vertical lines, like \(x = 4\), because:

• It helps locate exact x-values, providing a baseline from which vertical lines extend upwards and downwards.
• It allows the graphing of equations where the x-coordinate is constant.

In summary, the x-axis is a fundamental component in graphing, serving as a guide for accurately determining positions and understanding the layout of the coordinate plane. Knowing how to navigate the x-axis is vital for solving problems in coordinate geometry effectively.