Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Plot the points and find the slope of the line passing through the points. \(\left(-\frac{1}{3}, 1\right),\left(-\frac{2}{3}, \frac{5}{6}\right)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points \(\left(-\frac{1}{3}, 1\right)\), and \(\left(-\frac{2}{3}, \frac{5}{6}\right)\) is \(-\frac{1}{6}\).

Step 1: Identify the Coordinates

The coordinates of the two points to be plotted are \(\left(-\frac{1}{3}, 1\right)\) and \(\left(-\frac{2}{3}, \frac{5}{6}\right)\). So, \(x_1 = -\frac{1}{3}\), \(y_1 = 1\), \(x_2 = -\frac{2}{3}\), and \(y_2 = \frac{5}{6}\).

Step 2: Plot Points on a Graph

On a Cartesian plane, plot the first point at \(-\frac{1}{3}\) on the x-axis and \(1\) on the y-axis. Similarly, plot the second point at \(-\frac{2}{3}\) on the x-axis and \(\frac{5}{6}\) on the y-axis.

Step 3: Calculate the Slope

With the points plotted, the slope of the line passing through the points can be calculated using the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting with our values, we get \(m = \frac{\frac{5}{6} - 1}{-\frac{2}{3} - -\frac{1}{3}} = -\frac{1}{6}\).

Key Concepts

Coordinates

Coordinates are simply pairs of numbers that provide a precise position on a plane. Each coordinate pair consists of an

• x-coordinate: This represents the horizontal position on the plane, and
• y-coordinate: This denotes the vertical position on the plane.

These coordinates are usually written in the form \(x, y\), where "x" is the x-coordinate and "y" is the y-coordinate.
In the original exercise, we encountered two points with coordinates \(\left(-\frac{1}{3}, 1\right)\) and \(\left(-\frac{2}{3}, \frac{5}{6}\right)\). Here, "-\frac{1}{3}" and "-\frac{2}{3}" are the x-coordinates, and "1" and "\frac{5}{6}" are the y-coordinates.
This simple representation helps us locate specific points on the Cartesian Plane, which is crucial for further mathematical applications like finding the slope.

Cartesian Plane

Imagine a large sheet of paper divided into four parts by a big cross running through its center. This is what we visualize when thinking about the Cartesian Plane. It consists of:

• X-Axis: The horizontal line that runs side to side.
• Y-Axis: The vertical line that runs up and down.

The center where these axes meet is called the "origin," marked as (0, 0).
When you plot a point on this plane, you move horizontally to the x-coordinate and then vertically to the y-coordinate. For example, for the point \(\left(-\frac{1}{3}, 1\right)\), you move left to "-\frac{1}{3}" on the x-axis and up to "1" on the y-axis.
The Cartesian Plane allows us to see relationships between points, like the direction or slope of a line passing through them.

Slope Formula

The slope of a line indicates how steep the line is. It tells us how much the y-coordinate changes for a unit change in the x-coordinate. The slope formula, used to find the slope between two points, is written as: \[m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here,

• "\(m\)" represents the slope of the line,
• "\(y_2\)" and "\(y_1\)" are the y-coordinates of the two points, and
• "\(x_2\)" and "\(x_1\)" are the x-coordinates of the two points.

Using the example given \(\left(-\frac{1}{3}, 1\right)\) and \(\left(-\frac{2}{3}, \frac{5}{6}\right)\), we substitute the values: \(\frac{5}{6} - 1\) and \(-\frac{2}{3} - \left(-\frac{1}{3}\right)\), which simplifies to a slope of \(-\frac{1}{6}\).
Understanding the slope formula helps us not only to determine the tilt of the line but also to predict the direction in which the line extends as we move along the x-axis.