Problem 45 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. \((-5,0),(4,0)\)

Step-by-Step Solution

Answer

The slope of the line is 0.

Step 1: Understand the Slope Formula

The formula to find the slope \( m \) of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula calculates the change in \( y \) (rise) over the change in \( x \) (run).

Step 2: Identify the Coordinates

Plug in the given points \((-5,0)\) and \((4,0)\) into the formula. Here, \(x_1 = -5\), \(y_1 = 0\), \(x_2 = 4\), \(y_2 = 0\).

Step 3: Substitute into the Slope Formula

Substitute the identified coordinates into the slope formula: \[ m = \frac{0 - 0}{4 - (-5)} \]

Step 4: Simplify the Expression

Simplify the expression: \( m = \frac{0}{4 + 5} = \frac{0}{9} = 0 \). Therefore, the slope of the line is 0.

Key Concepts

Slope Formula

The slope formula is a fundamental concept in mathematics when dealing with lines. It helps us determine how steep a line is. In more technical terms, the slope gives us the rate at which one variable changes in relation to another. The formula often used is:

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

This equation simply measures the change in the "vertical direction" (known as the "rise") divided by the change in the "horizontal direction" (called the "run").
It's important to think of the slope as a ratio, telling us how much the line goes up for every unit it goes across. A positive slope indicates the line rises as it moves from left to right, while a negative slope shows it descends. A zero slope often indicates a flat line, and a "not defined" slope is usual for vertical lines. Understanding and using the slope formula can greatly assist in graphing lines and understanding their behavior.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using a coordinate plane to represent geometric shapes and solve problems related to them. In this context, it's crucial for understanding the position and movement of points, lines, and curves through a standardized graphical system.
Each point in coordinate geometry is represented by an ordered pair

• \((x, y)\)

where \(x\) is the distance along the horizontal axis and \(y\) is the distance along the vertical axis. This system allows us to analyze the geometric properties of shapes and to perform various calculations using the coordinates. For instance, we can easily calculate the distance between points or the midpoint of a line segment.
Coordinate geometry provides a powerful visual and algebraic method to prove geometric theorems and find solutions to problems involving shapes and lines in a plane. It forms a bridge between algebra and geometry, offering a deeper understanding of both.

Horizontal Line

A horizontal line is a type of line where all the points share the same \( y \)-coordinate. When plotting this on a graph, the result is a line that extends perfectly flat from left to right.
Such lines are characterized by having a slope of 0 because there is no change in the vertical direction (or "rise") no matter how much you move along the horizontal direction ("run"). Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), when \( y_2 = y_1 \), the numerator becomes 0, resulting in a slope of 0.

• Example: Between two points \((-5, 0)\) and \((4, 0)\), the slope is 0 as calculated earlier.

Horizontal lines are often associated with concepts like equilibrium or constancy in various fields such as mathematics, economics, and physics. They're easy to spot and helpful for visually interpreting data that doesn't change over an interval.