Problem 33 - Algebra 1: Concepts and Skills

Question

ZERO OR UNDEFINED SLOPE Determine whether the slope is zero, undefined, or neither. \((8,7)\) and \((14,1)\)

Step-by-Step Solution

Answer

The slope is neither zero nor undefined. It's -1.

Step 1: Identify the coordinates

The first point is (8,7) which is \((x_1,y_1)\) and the second point is (14,1) which is \((x_2,y_2)\).

Step 2: Calculate the changes in the coordinates

The change in x, denoted as \( \Delta x\) or \(x_2 - x_1\), is calculated as \(14 - 8 = 6\). Similarly, the change in y, denoted as \( \Delta y\) or \(y_2 - y_1\), is calculated as \(1 - 7 = -6\).

Step 3: Compute the slope

The slope (m) is the ratio between the vertical change and the horizontal change, or \( \Delta y / \Delta x\). Therefore \(m = -6/6 = -1\).

Key Concepts

Understanding Coordinates

Coordinates are ordered pairs that help us locate points on a grid, like spots on a map. Each pair is written as \((x, y)\), where:

• The first number \(x\) represents the horizontal position, telling us how far left or right the point is.
• The second number \(y\) indicates the vertical position, showing how far up or down the point is.

For example, in the pair \((8, 7)\), the point is 8 units to the right and 7 units up from the origin of the grid.In our exercise, the coordinates \((8,7)\) and \((14,1)\) mark two specific points on the plane.Recognizing these coordinates is the first step to understanding the changes in the values and calculating the slope.

Measuring Change in x

The change in x, often symbolized as \( \Delta x \), represents the horizontal shift between two points.Simply put, it is the difference in the x-values of the points. This indicates how far left or right one point is from another.To find the change in x, you subtract the x-coordinate of the first point from the x-coordinate of the second point.
In our example:

• Start with the x-values of the coordinates: 14 and 8.
• Calculate \( \Delta x = 14 - 8 = 6 \).

This tells us there is a horizontal shift of 6 units to the right from the first point to the second.

Measuring Change in y

The change in y, denoted as \( \Delta y \), defines the vertical shift between two points on a plane.This is the difference in the y-coordinates of our points and reveals how far one point moves up or down in relation to another.To determine the change in y, simply subtract the y-coordinate of the first point from the y-coordinate of the second point.
In this particular case:

• Take the y-values from the coordinates: 1 and 7.
• Compute \( \Delta y = 1 - 7 = -6 \).

Here, the negative sign indicates that there is a downward movement of 6 units, as the second point is lower than the first one.

Executing Slope Calculation

Slope is a measure of how steep a line connecting two points is on a graph. It tells us the direction and the incline of a line, which can be calculated using the formula:\[m = \frac{\Delta y}{\Delta x}\]where \( m \) represents the slope of the line.The formula involves two key components:

• \( \Delta y \): the difference in y-values (vertical change).
• \( \Delta x \): the difference in x-values (horizontal change).

In our exercise, the slope is calculated as:\[m = \frac{-6}{6} = -1\]This result shows a downward slope, meaning the line leans downwards from left to right.A slope of -1 indicates that for each step right (or unit increase in x), the line drops one unit down.