Problem 8 - College Algebra Essentials

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. $$(6,-4) \text { and }(4,-2)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (6, -4) and (4, -2) is -1. Thus, the line is falling.

Step 1: Identify Points

First, identify the points in question. These are (6,-4) and (4,-2). Assign the points to \(x_1\), \(y_1\) and \(x_2\), \(y_2\). So, \(x_1 = 6\), \(y_1 = -4\), \(x_2 = 4\), and \(y_2 = -2\).

Step 2: Calculate Slope

Use the formula \((y_2 - y_1) / (x_2 - x_1)\) to find the slope. Substitute the coordinates into the formula to get \((-2 - (-4)) / (4 - 6)\) which simplifies to \(2 / -2\). The slope is thus -1.

Step 3: Analyze Slope

The slope of -1 means for every 1 unit the x-coordinate increases, the y-coordinate decreases by 1 unit, so the line is falling from left to right. A line with a negative slope is always falling, so this line would neither be horizontal nor vertical, it is a falling line.

Key Concepts

Coordinate Geometry

Coordinate geometry or analytic geometry combines algebra and geometry to describe and analyze points, lines, and shapes using a coordinate system. This allows us to study geometric figures using algebraic equations.

For lines specifically, we use ordered pairs—commonly known as coordinates—in a plane. Each point is usually represented as \((x, y)\), where \(x\) is the horizontal position and \(y\) is the vertical position. The distance between points and their alignment, such as being on the same line, can be described logically using coordinate geometry.

• A major advantage of coordinate geometry is it allows abstract algebraic processes to yield tangible geometric results.
• The concept is foundational, as it helps in determining relationships and shapes through formulas and calculations.

Linear Equations

Linear equations are mathematical expressions that create straight lines when plotted on a coordinate plane. A typical linear equation is represented as \(y = mx + c\), where \(m\) represents the slope of the line, and \(c\) is the y-intercept, or the point where the line crosses the y-axis.

The slope \(m\) is crucial, as it determines the line's inclination relative to the x-axis. It can be positive, negative, zero, or undefined:

• Positive slope: The line rises as you move from left to right.
• Negative slope: The line falls as you move from left to right, as with the example in the exercise.
• Zero slope: The line is horizontal.
• Undefined slope: The line is vertical.

Linear equations are versatile tools in various fields. They help solve real-world problems where relationships between quantities need to be analyzed.

By understanding the components of linear equations, one can efficiently graph and interpret lines in the coordinate plane.

Graphing Lines

To understand graphing lines, it's essential to grasp how linear equations translate into visual representations on a coordinate plane. Graphing a line involves plotting points that satisfy the linear equation \(y = mx + c\), and then connecting these points to reveal the line.

The slope \(m\) indicates the tilt or angle of the line:

• For a slope of 1, the line rises at a 45-degree angle.
• For a negative slope like -1, as seen in our example, the line falls at a symmetrical angle.

The y-intercept \(c\) marks the starting point on the y-axis for the graph. This is where the line intersects the vertical axis.

Steps to Graph a Line:

• Identify the y-intercept on the graph and plot it.
• Use the slope to determine the next point. From the y-intercept, you "rise" and "run" according to the slope's ratio, moving up/down and left/right.
• Draw a line through the two points and extend it across the graph.

Graphing not only provides a visual understanding of equations but also aids in predicting and analyzing trends and relationships in data effectively.