Problem 45 - A Graphical Approach to Precalculus with Limits

Question

Find the slope (if defined) of the line that passes through the given points. $$\left(\frac{2}{3}, 9\right) \text { and }\left(\frac{1}{2}, 9\right)$$

Step-by-Step Solution

Answer

The slope is 0.

Step 1: Identify the Points

We are given the points \( \left(\frac{2}{3}, 9\right) \) and \( \left(\frac{1}{2}, 9\right) \). Let's denote them as \( (x_1, y_1) \) and \( (x_2, y_2) \) respectively.

Step 2: Write the Formula for the Slope of a Line

The formula to find the slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 3: Substitute the Point Coordinates into the Slope Formula

Using the points we have identified, substitute \( x_1 = \frac{2}{3}, y_1 = 9, x_2 = \frac{1}{2}, y_2 = 9 \) into the slope formula: \[ m = \frac{9 - 9}{\frac{1}{2} - \frac{2}{3}} \]

Step 4: Simplify the Expression

The numerator of the slope is \( 9 - 9 = 0 \). Thus, the slope expression becomes: \[ m = \frac{0}{\frac{1}{2} - \frac{2}{3}} \]Since the numerator is zero, the overall value for the slope \( m \) is zero, regardless of the denominator.

Step 5: Determine if Slope is Defined

Since the numerator is 0 and not both points have the same x-coordinates, the slope, which is 0, is well-defined. This indicates a horizontal line.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a powerful mathematical tool that allows us to describe geometric shapes using algebra. This integration helps in understanding the position of points, lines, and planes. Using a coordinate system, you can precisely express the location of any point.
The Cartesian coordinate system is the most commonly used, where each point can be described with an ordered pair \(x, y\). Here, \(x\) and \(y\) represent the horizontal and vertical distances from the origin \(0, 0\). When two points are given, such as \(\left(\frac{2}{3}, 9\right)\) and \(\left(\frac{1}{2}, 9\right)\), coordinate geometry helps us explore the relationship between these points and the line that passes through them.

• Points: Each point is defined by its coordinates.
• Lines: The position of a line can be studied by examining its slope (steepness) and intercepts.

Coordinate geometry not only facilitates sketching shapes but also solving complex mathematical problems with ease.

Linear Equations

Linear equations form the backbone of coordinate geometry. A linear equation describes a line in an algebraic form. It usually appears as \(y = mx + c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. The slope \(m\) signifies how steep the line is, and it is calculated based on the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
This expression shows how the vertical change between two points relates to the horizontal change.
For example, in the given problem, the slope formula simplifies to \(m = 0\). This indicates that our line is perfectly horizontal.

• Slope Zero: A zero slope suggests no rise or fall; the line runs parallel to the \(x\)-axis.
• Equation of the Line: With a slope of zero, the linear equation is simply \(y = c\), where \(c\) is the constant y-coordinate.

Linear equations are a gateway to modeling real-world scenarios, providing insights into changes and tendencies.

Horizontal Line

A horizontal line is special in coordinate geometry because it remains constant in the vertical direction. This occurs when the slope is zero. For any horizontal line passing through points like \(\left(\frac{2}{3}, 9\right)\) and \(\left(\frac{1}{2}, 9\right)\), the important aspect is that the y-coordinates are identical.
This leads to some characteristics worth noting:

• Constant y-value: A horizontal line does not change height; it retains the same y-value along the entire stretch.
• Equation Form: For our example, the equation is simply \(y = 9\), indicating that no matter the x-value, the line maintains this constant.
• No Vertical Change: The absence of vertical change results in a slope of zero. Mathematically, any zero numerator in the slope formula implies this kind of line.

Horizontal lines are easy to identify and graph, making them a fundamental concept in learning coordinate systems and linear equations.