Problem 30 - Algebra 1: Concepts and Skills

Question

ZERO OR UNDEFINED SLOPE Determine whether the slope is zero, undefined, or neither. $$ (1,2) \text { and }(1,6) $$

Step-by-Step Solution

Answer

The slope of the line passing through the points (1,2) and (1,6) is undefined.

Step 1: Identify the coordinates

Identify the coordinates of the two given points. The first point is (1,2) and the second point is (1,6) where the x-coordinate of both points is 1 and the y-coordinates are 2 and 6, respectively.

Step 2: Calculate the slope

Calculate the slope using the formula \((y_2 - y_1) / (x_2 - x_1)\). Substitute the coordinates into the slope formula where \(x_1 = 1\), \(y_1 = 2\), \(x_2 = 1\), and \(y_2 = 6\). By substitution, the formula becomes \((6 - 2) / (1 - 1)\).

Step 3: Analyze the result

Recognize that any number divided by zero is undefined. So, the slope of the line that passes through the points (1,2) and (1,6) is undefined.

Key Concepts

Undefined Slope

When it comes to slopes in algebra, there is a special scenario known as an "undefined slope." This happens when you try to divide a number by zero during slope calculation.

A line that rises vertically, such as the one that connects points like (1, 2) and (1, 6), has coordinates where the x-values are the same.
This means the line does not "run" or move horizontally at all.

• A vertical line's slope is considered undefined because slope represents a ratio of vertical change to horizontal change.
• Since the horizontal change is zero for vertical lines, this causes the slope formula to attempt dividing by zero.

In simpler terms, if the x-coordinates of two points are identical, the slope is undefined, making the line perfectly vertical.

Slope Calculation

Calculating the slope of a line is an essential skill in coordinate geometry.
The formula for calculating the slope between two points, extbf{(x_1, y_1)} and extbf{(x_2, y_2)}, is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
This formula helps us understand how steep a line is, or how much it tilts.

• The "rise" is calculated by subtracting the first y-coordinate (y_1) from the second y-coordinate (y_2).
• The "run" is calculated by subtracting the first x-coordinate (x_1) from the second x-coordinate (x_2).
• The slope (m) is the quotient of the rise over the run.

Remembering that when the denominator is zero, the slope becomes undefined is key.
Always check both the rise and the run carefully to avoid miscalculations.

Coordinates Analysis

Coordinates form the backbone of any slope calculation and are vital for understanding geometry on the coordinate plane.
Each point is represented by a pair (x, y) which shows its specific location on the grid.

• The x-value shows the position left or right along the horizontal axis.
• The y-value indicates the position up or down along the vertical axis.

Analyzing coordinates involves comparing these values between points to determine differences.
In our example, we see that both points (1,2) and (1,6) share the same x-value, signaling no change horizontally.
More generally, knowing how to read and analyze coordinates is crucial for tasks beyond just calculating slope, such as graphing lines and determining the relation between multiple points on a plane.