Problem 19 - Thinking Mathematically

Question

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((5,1)\) and \((-2,1)\)

Step-by-Step Solution

Answer

The slope of the line passing through points (5,1) and (-2,1) is 0, indicating a horizontal line.

Step 1: Identify the coordinates

Given two points, point A is \( (x1, y1) = (5,1) \) and point B is \( (x2, y2) = (-2,1) \).

Step 2: Calculate the Slope

Using the slope-formula \(m = \frac{y2 - y1}{x2 - x1}\), it gives \(m = \frac{1 - 1}{-2 - 5}\).

Step 3: Simplify the equation

On simplification, \(m = \frac{0}{-7}\) gives slope \(m = 0\). Because the denominator is not zero, the slope is defined.

Step 4: Analyze the slope

Slope 0 corresponds to a horizontal line. Hence, the line does not rise or fall.

Key Concepts

Coordinate Geometry

Coordinate geometry, often known as analytic geometry, is a fascinating field where geometric figures are studied and analyzed using a coordinate system. It helps us to quantify and study shapes, sizes, and even the relative positions of figures. Using coordinate geometry, you can represent points in a plane through a set of numbers. This is typically done using pairs, referred to as Cartesian coordinates, which consist of two values: the x-coordinate (horizontal position) and the y-coordinate (vertical position).

In the problem we encountered, there are two points:

• Point A:
  • Coordinates:
    • X: 5, Y: 1
• Point B:
  • Coordinates:
    • X: -2, Y: 1

Both points share the same y-coordinate (1). This gives us the first hint about the nature of the line passed through these points, which we explore through slope calculations. The beauty of coordinate geometry is how it connects algebra and geometry, giving us a powerful tool to analyze and understand space.

Slope Formula

The slope formula is a key element in understanding the inclination of lines, a crucial concept in algebra and calculus. Slope, often denoted as ‘m’, is a measure of how steep a line is. It’s calculated by determining the ratio of the vertical change to the horizontal change between two distinct points on a line.

• Slope Formula:
  • \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

In our exercise, applying the formula to points

• the y-coordinates are the same: (1 - 1)

produces a zero difference in height, so the result of the numerator becomes zero. The denominator,

• (-2 - 5),

gives us -7 after subtracting the x-coordinates. Simplifying gives us:

• Slope \( m = \frac{0}{-7} = 0 \).

This shows us that not only does the line not change in y-value, but also confirms that any horizontal line will always have a slope of zero. Understanding this fundamental slope calculation helps us determine whether a line is inclined upwards, downwards, or remains flat (horizontal).

Horizontal Line

A horizontal line is a fascinating and simple concept in geometry. It’s characterized by having a constant y-value, which means it doesn't rise as it moves from left to right, it remains constant in height. In this particular problem, we discovered the line to be horizontal by calculating the slope to be zero.

• The characteristics of horizontal lines include:
  • Slope: 0
  • Equation: Generally written as \(y = c\), where \(c\) is the y-intercept or constant of the line.

Horizontal lines indicate that there is no rise or fall, and hence they are level.

In our example, with both points having a y-coordinate of 1, we know the entire line sits on this line, confirming our calculations and showing no rise or fall between the points given. Regardless of the x-values or how far apart horizontally the points may lie, the line itself is flat on the plane. This unique trait makes horizontal lines an easy type to identify once you’re equipped with the understanding of coordinate geometry and slope calculations.