Problem 38 - Algebra 1

Question

Decide whether the line is horizontal or vertical. Then graph the line. \(x=-5\)

Step-by-Step Solution

Answer

The given line is a vertical line. To graph it, a straight line should be drawn at \(x = -5\).

Step 1: Identify the Type of Line

The equation given is \(x = -5\), which is of the form \(x = c\). This means that we're dealing with a vertical line, where every point on this line has an x-coordinate of -5.

Step 2: Graph the Line

To graph this line, a straight vertical line needs to be drawn at \(x = -5\) on a coordinate plane. It doesn't matter what the y-coordinates are because they can take any value; the x-coordinate will always be -5.

Key Concepts

Understanding Vertical Lines

Vertical lines are a fundamental concept in graphing. They have an equation of the form \(x = c\), where \(c\) is a constant. This means that:

• Every point on the line has the same x-value, \(c\).
• The y-values can be anything—they are unrestricted.

Think of vertical lines as boundaries on the coordinate plane. Imagine a line drawn straight up and down at a particular x-coordinate. For example, if the line is \(x = -5\), every point along this line has an x-coordinate of -5. It's like an invisible barrier that runs parallel to the y-axis. When you see the equation \(x = -5\), you can be sure it represents a vertical line.

Exploring the Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can graph points, lines, and curves. It is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical). Here's what you need to know:

• X-Axis: Divides the plane into top and bottom halves.
• Y-Axis: Divides the plane into left and right halves.
• Quadrants: Numbered counter-clockwise starting from the upper right quadrant.

Each point on the coordinate plane is identified by a pair of numbers, \((x, y)\), where \(x\) is the horizontal position and \(y\) is the vertical position. When graphing a line like \(x = -5\), you locate -5 on the x-axis and draw a straight line parallel to the y-axis. It's essential to grasp the layout of the coordinate plane to effectively graph and interpret lines.

Introduction to Equations of Lines

Equations of lines are mathematical representations of straight lines on the coordinate plane. They provide a way to describe the characteristics of a line:

• The general form for a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
• However, vertical lines deviate from this form. They use the equation \(x = c\), representing a vertical line at the position \(c\) on the x-axis.

This distinction is crucial as vertical lines do not have a slope as defined in the usual sense of rise over run. Their slope is considered undefined because a vertical change without horizontal movement results in division by zero. Recognizing the type of line based on its equation allows you to accurately graph and understand its behavior in the coordinate system.