Problem 35 - Precalculus with Limits

Question

In Exercises 29-40, plot the points and find the slope of the line passing through the pair of points. \((-6, -1)\), \((-6, 4)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points \(-6, -1\) and \(-6, 4\) is undefined, indicating a vertical line.

Step 1: Plot the Points

To begin, plot the points \(-6, -1\) and \(-6, 4\) on a coordinate grid. The two points would be placed vertically in line with each other because they share the same x-coordinate (which is -6).

Step 2: Apply the Slope Formula

To find the slope, apply the formula: slope = \(\frac{(y_2 - y_1)}{(x_2 - x_1)}\). Substituting \(-6, -1\) for \((x_1, y_1)\) and \(-6, 4\) for \((x_2, y_2)\) we get: \(\frac{(4 - (-1))}{(-6 - (-6))}\).

Step 3: Solve for the Slope

Solving, we have \(\frac{5}{0}\), which is undefined due to division by zero. Therefore, the slope of the line is undefined, meaning that the line is vertical.

Key Concepts

Coordinate Geometry

Coordinate geometry, often called analytic geometry, is a mathematical study that utilizes a coordinate system to explore geometric properties. This allows us to describe geometric shapes in a numerical way, making it easier to analyze and solve problems. Points on a plane are represented using pairs of numbers, known as coordinates. These coordinate pairs are structured as \(x, y\), where \(x\) is the horizontal distance from the origin and \(y\) is the vertical distance.

In this specific exercise, coordinate geometry helps us by providing a framework to plot points on a grid. This grid is defined by two perpendicular lines (often called axes) labeled as the x-axis (horizontal) and y-axis (vertical). Once these points are plotted, we can evaluate the geometric relationship between them, such as determining the slope of the line connecting them. This involves algebraic computations using coordinates to identify characteristics like slope, length, and area.

Plotting Points

Plotting points is a fundamental part of coordinate geometry and is crucial for visualizing the relationships between different points. To plot a point, you need its coordinates \(x, y\). The \(x\) coordinate tells you how far to move horizontally from the origin (0,0), and the \(y\) coordinate tells you how far to move vertically.

In our exercise, the points are \((-6, -1)\) and \((-6, 4)\). For each point:

• Locate -6 on the x-axis, because both points share the same x-coordinate.
• Then, move up to -1 for the first point and up to 4 for the second point on the y-axis.
• Place a dot at each position to mark the points.

Since both points have the same x-coordinate, they line up vertically, directly above or below each other on the grid. This tells us that there's a vertical line passing through both points.

Vertical Line

Vertical lines have unique properties that distinguish them within coordinate geometry. A vertical line runs parallel to the y-axis and is characterized by having an undefined slope. The reason for this undefined slope emerges from the slope formula: \[ \text{slope} = \frac{(y_2 - y_1)}{(x_2 - x_1)} \]In vertical lines, the x-coordinates of all points are identical, leading to zero as the denominator in the formula. As division by zero is not possible, we say the slope is undefined.

For the points \((-6, -1)\) and \((-6, 4)\), the denominator becomes \( -6 - (-6) = 0\), confirming the vertical nature of the line.

Recognizing vertical lines is important because it affects how we perceive motion and relationships in space. Vertical lines visually appear as straight up-and-down connections on a graph, underpinning concepts such as parallelism and symmetry in geometry.