Problem 8 - College Algebra

Question

Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. $$(6,-4)\( and \)(4,-2)$$

Step-by-Step Solution

Answer

The slope of the line that passes through the points (6,-4) and (4,-2) is -1. The line falls.

Step 1: Identify the given points

The given points are (6,-4) and (4,-2). Let's denote them as: point 1, \( P_1(6,-4) \) where \( x_1 = 6 \) and \( y_1 = -4 \); and point 2, \( P_2(4,-2) \) where \( x_2 = 4 \) and \( y_2 = -2 \).

Step 2: Use the slope formula

The slope of a line is given by the formula \( m = (y_2 - y_1) / (x_2 - x_1) \). Now, substitute the values of \( x_1, y_1, x_2, y_2 \) in the formula to find the slope: \( m = (-2 - (-4)) / (4 - 6) \).

Step 3: Simplify to find the slope

Simplify the above expression to find the value of \( m \): \( m = (-2 + 4) / (4 - 6) \) which simplifies to \( m = 2 / -2 = -1 \).

Step 4: Determine the direction of the line

The slope is -1 which is a negative value. Therefore, the line falls (since a negative slope indicates a line that falls from left to right).

Key Concepts

Linear Equations

Linear equations are mathematical expressions used to describe a straight line on a coordinate plane. They usually take the form \( y = mx + b \), where:\

• \( m \) is the slope of the line, representing how steep the line is, and
• \( b \) is the y-intercept, indicating where the line crosses the y-axis.

In the solution you encountered, the focus was on determining the slope \( m \). This calculation is crucial as it defines the direction and steepness of the line. Any two points \((x_1, y_1)\) and \((x_2, y_2)\) can be used to find \( m \) using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Understanding this formula is key to solving problems related to linear equations. Once you find \( m \), you can use it to write the equation of the line or determine its behavior, such as increasing, decreasing, horizontal or vertical.

Coordinate Geometry

Coordinate geometry involves using algebraic expressions to solve geometric problems by placing them in a coordinate plane. This allows for a visual interpretation of relationships between points, lines, and shapes.
The solution process given above makes use of coordinate geometry to find and interpret the slope of the line passing through two points. In a coordinate plane:

• Each point is represented by a pair of coordinates \((x, y)\).
• The distance or difference between these coordinates helps calculate the slope.
• By plotting the line between points \((6,-4)\) and \((4,-2)\), you can visually observe that it falls as you go from left to right.

The coordinate system acts as a bridge between algebraic concepts and their graphical representation, allowing you to see how mathematical rules apply to spatial relations.

Negative Slope

A negative slope signifies that as you move from left to right across a graph, the line moves downward. This is in contrast to a positive slope, where the line would rise. The slope is calculated by the change in \( y \) over the change in \( x \) between two points.
In the exercise, the slope of \(-1\) was derived from: \[ m = \frac{-2 - (-4)}{4 - 6} = \frac{2}{-2} = -1 \] which revealed a negative slope.

• This tells us that the line passes through the coordinates, falling from \(P_1\) to \(P_2\).
• Such a slope often appears in contexts where there is a decrease or reduction, such as in decreasing temperature over time.
• The sign of the slope alone gives valuable information about the direction of the relation depicted by the line.

Understanding the concept of a negative slope helps in predicting and analyzing real-world scenarios where one variable decreases as another increases.