Problem 23 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the distinct lines through each pair of points are parallel. $$(-2,0)\( and \)(0,6) ;(1,8)\( and \)(0,5)$$

Step-by-Step Solution

Answer

The lines that pass through the point pairs (-2,0) and (0,6) and through (1,8) and (0,5) are indeed parallel because they both have the same slope of 3.

Step 1: Finding the Slope of the First Line

We will use the slope formula \((y_2 - y_1) / (x_2 - x_1)\) on the coordinates (-2,0) and (0,6) to find the slope of the first line. The slope will be \((6 - 0) / (0 - (-2)) = 3\)

Step 2: Finding the Slope of the Second Line

Using the slope formula again, we calculate the slope of the line through the points (1,8) and (0,5). The slope will be \((8 - 5) / (1 - 0) = 3\)

Step 3: Comparing the Slopes

Lastly, compare the calculated slopes. Those are equal, therefore the lines are parallel

Key Concepts

Slope Formula

The slope of a line is a measure of its steepness and direction. To calculate the slope, we use the slope formula, which is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points on the line. The numerator represents the change in the y-coordinates, while the denominator represents the change in the x-coordinates. This formula provides an intuitive understanding of how steep or flat a line is.

• If the slope \(m\) is positive, the line ascends from left to right.
• If it is negative, the line descends from left to right.
• If the slope is zero, the line is horizontal.
• An undefined slope indicates a vertical line.

The slope formula is central to determining the relationship between lines, such as parallel or perpendicular alignment.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves the study of geometry using a coordinate system. In this case, the Cartesian coordinate system allows us to represent geometric shapes and concepts algebraically. This makes it possible to solve geometric problems through algebraic methods.
When dealing with points in a plane, each point has its own coordinates, denoted as \( (x, y) \). Both coordinate points provided can describe a line when connected. Understanding the slope and position of these points is crucial in defining the characteristics of the line.
Using coordinates:

• We can determine the distance between two points using the distance formula.
• Find the slope of a line to understand its direction.
• Analyze the intersection or relationships between multiple lines, such as parallelism or perpendicularity.

This foundational understanding sets the stage for solving complex geometry problems algebraically.

Line Equations

Line equations are mathematical expressions that describe a line in terms of its slope and a point. The most common form is the slope-intercept form, given as: \[ y = mx + c \] In this equation, \(m\) represents the slope of the line, and \(c\) is the y-intercept, which is the point where the line crosses the y-axis. The slope-intercept form is particularly useful for quickly sketching a line and understanding its behavior.
Beyond this, there’s the point-slope form, given by: \[ y - y_1 = m(x - x_1) \] This form is ideal when you know a line’s slope and one point on the line.
Equations of lines help in:

• Predicting or defining a line's path across a plane.
• Determining intersections between different lines.
• Identifying features like parallelism by comparing slopes.

These equations bridge geometric concepts with algebraic operations, creating a powerful toolset for studying linear relationships.