Problem 24 - Algebra 2

Question

Graph the line passing through the given point with the given slope. \((-2,-3),\) undefined

Step-by-Step Solution

Answer

The line through (-2,-3) with an undefined slope is vertical: x = -2.

Step 1: Identify the Characteristics of an Undefined Slope

An undefined slope means the line is vertical. This is because the change in the x-values is zero causing the slope calculation to become undefined. For a vertical line, all points on the line have the same x-coordinate.

Step 2: Use the Given Point to Determine the Line's Equation

Since the line is vertical and must pass through the point (-2,-3), every point on this line will have an x-coordinate of -2. Therefore, the equation of the line is given by x = -2.

Step 3: Graph the Vertical Line on a Coordinate Plane

To graph the vertical line, locate the point (-2, -3) on the coordinate plane. Then draw a straight vertical line through this point, parallel to the y-axis. This line should extend infinitely in both upward and downward directions.

Key Concepts

undefined slope

When a line has an undefined slope, it means the line is oriented in a specific way: vertically.
Let's think about what a slope is first. Slope usually represents how steep a line is, calculated as the 'rise over run', or the change in the y-coordinates versus the change in the x-coordinates.
However, for a vertical line, the change in x-coordinates is zero because the line goes straight up and down.
This causes a division by zero in the slope formula, which results in an undefined value. Therefore, such lines are called vertical and have undefined slopes.

• Vertical lines go up and down.
• They don't tilt left or right, hence the term 'undefined slope'.

coordinate plane

The coordinate plane is like a map you use to find a location. It consists of two number lines that cross each other at right angles.
The horizontal line is called the x-axis, and the vertical line is the y-axis. The point where they intersect is known as the origin with coordinates (0, 0).
Each point on the coordinate plane is identified by a pair of numbers written as (x, y):

• The first number is the x-coordinate.
• The second number is the y-coordinate.

In our task involving the point (-2, -3), the line should be drawn on this coordinate map, helping visualize how lines and points relate to each other.

line equation

The line equation describes all the points that form the line on a coordinate plane. Usually written in the form of y = mx + b, where m is the slope and b the y-intercept.
However, vertical lines are special since their slopes are undefined. For these lines, the equation simplifies significantly because x is constant for all points on the line.
A vertical line only has an x-term, written as x = constant. In our example, x = -2, which defines the location of the line along the x-axis.

• This equation means the line passes through every point where x = -2.
• No y-term is needed as y-values can be any number.

x-coordinate

The x-coordinate is crucial in locating points on a coordinate plane. It's always the first number in the coordinate pair (x, y).
It signals how far a point is from the vertical y-axis to the left if negative and to the right if positive.
In vertical lines, like the one we are examining, the x-coordinate remains consistent. All points will share the same x-coordinate.
For instance, the line passing through the point (-2, -3) will maintain x = -2 uniformly:

• This means every point on the line is directly above or below (-2, -3).
• The y-coordinates change, but the line stays parallel to the y-axis, creating a perfectly vertical path.

Understanding the x-coordinate's role helps in drawing and interpreting these vertical lines effectively.