Problem 25 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the distinct lines through each pair of points are parallel. $$(0,3)\( and \)(1,5) ;(-1,7)\( and \)(1,10)$$

Step-by-Step Solution

Answer

The two lines are not parallel because their slopes are not equal. The slope of the line passing through points (0,3) and (1,5) is 2, while the slope of the line passing through points (-1,7) and (1,10) is 1.5.

Step 1: Calculate the slope for the first pair of points

The first pair of points is (0,3) and (1,5). The slope between these points can be calculated as follows: Using the slope formula \[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} \] we get \[ m_1 = \frac{5 - 3}{1 - 0} = 2 \]

Step 2: Calculate the slope for the second pair of points

The second pair of points is (-1,7) and (1,10). The slope between these points can also be calculated with the slope formula \[ m_2 = \frac{y_2 - y_1}{x_2 - x_1} \]giving us\[ m2 = \frac{10 - 7}{1 - (-1)} = 1.5 \]

Step 3: Compare the two slopes

Now that we have both slopes, we can compare them. If the slopes were equal, the lines would be parallel. However, m1 is not equal to m2, so the two lines are not parallel.

Key Concepts

Slope Formula

The slope formula is essential in understanding how steep a line is when it's plotted on a graph. In mathematics, the slope of a line is often represented by the letter \( m \). It's a measure of how much the \( y \)-value changes per unit increase in the \( x \)-value. The formula is written as:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are two distinct points on a line.
The difference in \( y \)-values (\( y_2 - y_1 \)) represents how much the line rises or falls (known as "rise"), while the difference in \( x \)-values (\( x_2 - x_1 \)) represents how much the line moves horizontally (known as "run").
When determining if two lines are parallel, you compare their slopes.
If the slopes are identical, the lines will never meet, hence they are parallel. If they are different, as seen in our original exercise, then the lines will intersect at some point.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, bridges the gap between algebra and geometry. It uses a coordinate plane to graphically represent algebraic equations.
This powerful tool allows us to visualize geometric shapes and analyze their properties using algebraic equations.
The coordinate plane consists of two axes:

• The horizontal axis, known as the \( x \)-axis
• The vertical axis, known as the \( y \)-axis

Each point on this plane is defined by a pair of numbers \( (x, y) \), denoting its position relative to these axes.
Through coordinate geometry, we can calculate distances, identify midpoints, and in this context, determine the slope of a line connecting two points. The coordinates are vital because they allow us to apply concepts like the **slope formula** to judge relationships between lines, such as parallelism or perpendicularity.

Line Equation

In coordinate geometry, a line can be described using an equation. The most common form is the slope-intercept form:

• \( y = mx + b \)

where \( m \) is the slope, and \( b \) is the \( y \)-intercept (the point where the line crosses the \( y \)-axis).
This form makes it easy to graph a line and visualize its angle and position relative to the axes.To derive the line equation, we use a known point on the line in conjunction with the slope. Given the point \( (x_1, y_1) \) and the slope \( m \), the point-slope form of the equation is:

• \( y - y_1 = m(x - x_1) \)

You can transform this into the slope-intercept form to interpret graphical representations more easily.
Understanding the line equation allows us to describe and predict the behavior of linear graphs efficiently. It directly connects to the main task of determining if lines are parallel by comparing the slopes extracted from their equations.