Problem 9 - Precalculus Essentials

Question

Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. $$(5,3) \text { and }(5,-2)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (5,3) and (5,-2) is undefined and the line is vertical.

Step 1: Identifying the given points

Identify the given points. In this case, they are (5,3) and (5,-2). So, the first point, x1=5, y1=3. And the second point, x2=5, y2=-2.

Step 2: Calculate the Slope

The slope is calculated as (y2-y1)/(x2-x1). Here, substituting the values, we get (-2-3)/(5-5) = -5/0.

Step 3: Determine the Slope

However, division by zero is undefined in mathematics. Therefore, the slope of the line passing through the points is undefined.

Step 4: Description of the Line

Since the x-coordinates of both points are the same, it's clear that no matter what the value of y, x remains the same. This indicates that the line is vertical.

Key Concepts

Undefined Slope

The concept of slope is central in understanding the nature of a line in a coordinate plane. To find the slope between two points, you subtract the y-coordinates and divide the result by the difference in the x-coordinates. In formula terms, this is expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

However, what happens if the x-coordinates are the same? Suppose we have two points, such as (5,3) and (5,-2), which share the same x-coordinate. If you plug these into the formula, the denominator becomes zero, leading to division by zero, which is undefined in mathematics. This tells us that the slope doesn't exist in the usual sense for a line passing through vertical points.

• An undefined slope means that the line is perfectly vertical.
• Unlike other cases, there's no finite value that we can ascribe to the slope; it's a hint at a special kind of linear relation.

Vertical Line

A vertical line in the coordinate plane is a special type of line where all points have the same x-coordinate. This means the line does not 'rise' or 'fall'; it just continues up and down. In the example of the points (5,3) and (5,-2), if you plot these points on a graph, you will see a straight line going vertically through the x-value of 5.

Something crucial about vertical lines is that they show no horizontal change - that is, their x-coordinates are constant. With all x-values identical, this is why the slope formula results in division by zero.

Key notes about vertical lines include:

• They graph as vertical, straight lines parallel to the y-axis.
• They are often represented by an equation of the form \( x = \text{constant} \).
• They always have an undefined slope.

Points and Coordinates

When discussing lines on a plane, we refer to specific locations called points. Each point is defined by two numbers, known as coordinates, showing its exact position.

Each point has an x-coordinate, which tells how far left or right the point is from the vertical y-axis. It also has a y-coordinate, describing how far up or down it is from the horizontal x-axis. For example, if you have points (5,3) and (5,-2), this means:

• The first point is 5 units right on the x-axis, 3 units up on the y-axis.
• The second point is also 5 units right, but 2 units down.

When these points share the same x-coordinate but have different y-coordinates, like in our example, any line passing through them is vertical. This makes understanding the role of coordinates vital in predicting the type of line formed, notably when we encounter an undefined slope.

Knowing this helps us visualize and calculate the properties of lines simply and correctly.