Problem 11 - Elementary Algebra

Question

Find the slope of the line determined by each pair of points. $$(3,-4),(2,-4)$$

Step-by-Step Solution

Answer

The slope of the line is 0.

Step 1: Recognize the Slope Formula

The formula to find the slope (\(m\)) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Identify \((x_1, y_1)\) as (3, -4) and \((x_2, y_2)\) as (2, -4).

Step 2: Substitute the Values into the Slope Formula

Substitute the given points into the slope formula. You have:\[m = \frac{-4 - (-4)}{2 - 3}\]Simplify this expression by performing the arithmetic operations in the numerator and the denominator.

Step 3: Calculate the Numerator

Calculate the numerator by performing the subtraction:\[-4 - (-4) = -4 + 4 = 0\]So, the numerator is 0.

Step 4: Calculate the Denominator

Calculate the denominator by performing the subtraction:\[2 - 3 = -1\]So, the denominator is -1.

Step 5: Determine the Slope

Now, substitute the simplified numerator and denominator back into the formula:\[m = \frac{0}{-1} = 0\]Therefore, the slope of the line is 0, indicating a horizontal line.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that define a straight line. These equations are commonly written in the format of \(y = mx + c\), where \(m\) represents the slope of the line and \(c\) is the y-intercept. Understanding this format is crucial as it provides valuable information about how the line behaves in a coordinate plane.

The slope \(m\) indicates how steep the line is and which direction it moves. A positive slope means the line rises as it moves from left to right, while a negative slope suggests it falls. If \(m = 0\), the line is horizontal, and if \(c\) is zero, the line passes through the origin. The y-intercept \((c)\) tells us where the line crosses the y-axis.

• Equation of a straight line: \(y = mx + c\)
• Slope \(m\): Shows the inclination of the line
• Y-intercept \(c\): Where the line crosses the y-axis

Linear equations play a fundamental role in various fields, such as physics and economics, as they model relationships between two variables efficiently.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses coordinates to describe the position of a point, line, or other geometric objects in a plane. It combines algebra and geometry to provide a concrete representation of spatial forms.

This field uses the Cartesian coordinate system, which consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is known as the origin, represented by \((0,0)\). Points in this system are represented by pairs, like \((x, y)\).

Through coordinate geometry, you can:

• Calculate distances between points
• Find midpoints of line segments
• Determine slopes of lines

Understanding the concept of slope, as taught through coordinate geometry, helps in visualizing how and why lines appear as they do. Whether they are steep, gradual, or perfectly horizontal, slopes offer a foundational understanding of how lines are formed on a plane.

Horizontal Line

A horizontal line is a straight line with a constant value of y-coordinate, which means it runs parallel to the x-axis. In coordinate geometry, this type of line is critical as it signifies no vertical change; hence, the slope of a horizontal line is always zero.

To understand why the slope of a horizontal line is zero, consider two points on this line, say \( (x_1, y) \) and \( (x_2, y) \). Using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), for a horizontal line, this becomes \(m = \frac{y - y}{x_2 - x_1} = \frac{0}{x_2 - x_1}\). The numerator being zero results in the slope being zero, confirming the line does not rise or fall.

• Example of a horizontal line: \(y = c\)
• Slope of a horizontal line: Always zero
• Parallel to the x-axis

This concept is crucial when interpreting graphs as it indicates a constant relationship, with no change regardless of alterations in the x-variable. For example, if discussing speed, a horizontal line would represent a situation where there's no increase or decrease over time.