Problem 3 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 4 , find the slope of the line through the given points. $$ \left(\frac{1}{3}, \frac{1}{2}\right),\left(-\frac{5}{6}, \frac{2}{3}\right) $$

Step-by-Step Solution

Answer

The slope is \(-\frac{1}{7} \).

Step 1: Identify the coordinates

Given points are \(\frac{1}{3}, \frac{1}{2} \) and \(-\frac{5}{6}, \frac{2}{3} \). Label them as \(x_1 = \frac{1}{3}, y_1 = \frac{1}{2}, x_2 = -\frac{5}{6}, y_2 = \frac{2}{3} \).

Step 2: Recall the slope formula

The formula for the slope (m) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \(\frac{y_2 - y_1}{x_2 - x_1} \).

Step 3: Substitute the coordinates into the slope formula

Plug in the values \( (x_1 = \frac{1}{3}), (y_1 = \frac{1}{2}), (x_2 = -\frac{5}{6}), (y_2 = \frac{2}{3}) \) into the formula: \(\frac{\frac{2}{3} - \frac{1}{2}}{-\frac{5}{6} - \frac{1}{3}} \).

Step 4: Simplify the numerator

Find a common denominator for the numerator: \( \frac{2}{3} - \frac{1}{2} = \frac{4}{6} - \frac{3}{6} = \frac{1}{6} \).

Step 5: Simplify the denominator

Similarly, simplify the denominator with a common denominator: \(-\frac{5}{6} - \frac{1}{3} = -\frac{5}{6} - \frac{2}{6} = -\frac{7}{6} \).

Step 6: Divide the simplified numerator by the simplified denominator

Now, compute the slope: \(\frac{\frac{1}{6}}{-\frac{7}{6}} = \frac{1}{6} \times -\frac{6}{7} = -\frac{1}{7} \).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where we use a coordinate system to describe the position of points, lines, and shapes. In a coordinate system, each point is represented by an ordered pair \( (x, y) \), where \( x \) is the horizontal position, and \( y \) is the vertical position. This method allows us to analyze and solve geometric problems using algebraic formulas.
In this exercise, we are given two points: \( \left(\frac{1}{3}, \frac{1}{2}\right) \) and \( \left(-\frac{5}{6}, \frac{2}{3}\right) \). These points are located on a coordinate plane, and we need to determine the slope of the line that connects them.

Slope Formula

The slope formula helps us find the steepness or incline of a line between two points. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points. Mathematically, the slope \( m \) is expressed as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
In our given problem, we must identify the coordinates for the points provided: \( (x_1, y_1) = \left(\frac{1}{3}, \frac{1}{2}\right) \) and \( (x_2, y_2) = \left(-\frac{5}{6}, \frac{2}{3}\right) \). Substituting these values into the slope formula gives us:
\[ m = \frac{\frac{2}{3} - \frac{1}{2}}{-\frac{5}{6} - \frac{1}{3}} \]

Simplifying Fractions

When working with fractions, it is often necessary to simplify them to make calculations easier. Here, we must simplify both the numerator and the denominator separately before finding their quotient.
First, find a common denominator for the numerator:
\[ \frac{2}{3} - \frac{1}{2} = \frac{4}{6} - \frac{3}{6} = \frac{1}{6} \]
Next, simplify the denominator similarly:
\[ -\frac{5}{6} - \frac{1}{3} = -\frac{5}{6} - \frac{2}{6} = -\frac{7}{6} \]
Finally, divide the simplified numerator by the simplified denominator:
\[ \frac{\frac{1}{6}}{-\frac{7}{6}} = \frac{1}{6} \times -\frac{6}{7} = -\frac{1}{7} \]
Thus, the slope of the line through the points is \( -\frac{1}{7} \).