Problem 23 - Introductory Algebra for College Students

Question

In Exercises \(23-26,\) determine whether the distinct lines through each pair of points are parallel. $$(-2,0) \text { and }(0,6) ;(1,8) \text { and }(0,5)$$

Step-by-Step Solution

Answer

The two lines are not parallel because their slopes are not the same.

Step 1: Calculate the slope of the first line

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here \(x1=-2\), \(y1=0\), \(x2=0\), and \(y2=6\). By substituting these values, we find the slope \(m1 = \frac{6 - 0}{0 - -2} = 3\).

Step 2: Calculate the slope of the second line

Now, similarly calculate the slope of the second line using its two points which are \((x1'=1, y1'=8)\) and \((x2'=0, y2'=5)\). Here also apply the slope formula \(m' = \frac{y2' - y1'}{x2' - x1'}\). The slope \(m2 = \frac{5 - 8}{0 - 1} = -3\).

Step 3: Compare the slopes of the two lines

After computing the slopes \(m1\) and \(m2\), we now have to compare them to check if they are equal. If they are the same, then the lines are parallel. Substituting the obtained values we get \(3 = -3\). This statement is false, these lines are not parallel.

Key Concepts

Slope Formula

The slope formula is a fundamental concept in coordinate geometry. It is used to determine the steepness of a line in a plane. Specifically, the slope is the ratio of the vertical change to the horizontal change between two points on a line. This is expressed mathematically as:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where:

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of two distinct points on the line.
• \( m \) represents the slope.

This formula helps us understand how much \( y \) changes with respect to \( x \). A positive slope indicates the line rises from left to right, while a negative slope indicates it descends.Computing the slope is helpful in analyzing if lines are parallel. For example, lines are parallel if their slopes are equal.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that studies geometric figures using a coordinate system. In this system, points are defined by coordinates, usually in two-dimensional or three-dimensional space.In two-dimensional space:

• The location of a point is described using an ordered pair \((x, y)\).
• This pair represents its position relative to a reference, typically the origin \((0, 0)\) on a Cartesian plane.

Coordinate geometry allows for the algebraic manipulation of geometric figures. By using coordinates:

• We can calculate distances, slopes, and other properties.
• It aids in graphing linear equations and finding intersections between lines.

Understanding this topic paves the way for solving more complex geometric problems by converting them into algebraic equations.

Equation of a Line

The equation of a line is a way to represent the line algebraically and contains crucial information regarding its properties. There are several forms in which the equation of a line can be expressed, the most common being the slope-intercept form and the point-slope form.

• Slope-Intercept Form: This is expressed as \( y = mx + b \), where:
  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
• Point-Slope Form: Used when the slope \( m \) and a point \((x_1, y_1)\) are known, written as \[ y - y_1 = m(x - x_1) \]

The equations provide a way to define lines in relation to their slopes and positions. They are also essential in predicting geometrical behavior, such as whether two lines with equal slopes are parallel. A thorough understanding of the equation of a line helps solve a variety of problems not only in math but also in fields such as physics and engineering.