Problem 31 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the lines through each pair of points are parallel, perpendicular, or neither. $$(-2,-5)\( and \)(3,10) ;(-1,-9)\( and \)(4,6)$$$

Step-by-Step Solution

Answer

The lines through the given pairs of points are parallel.

Step 1: Calculate the slope of the first line

The points for the first line are (-2,-5) and (3,10). Using the formula for slope, \((y2 - y1) / (x2 - x1)\), the slope (m1) can be calculated as (10 - (-5)) / (3 - (-2)) = 15/5 = 3.

Step 2: Calculate the slope of the second line

The points for the second line are (-1,-9) and (4,6). Applying the same slope formula, the slope (m2) is calculated as (6 - (-9)) / (4 - (-1)) = 15/5 = 3.

Step 3: Comparison of slopes

Inspect the calculated slopes (m1 & m2) of the two lines to determine the relationship between the lines. As m1 equals m2, the two lines are parallel to each other.

Key Concepts

Slope Calculation

Slope calculation is a fundamental aspect of coordinate geometry, helping us understand the steepness or inclination of a line. Calculating the slope of a line, given two points, involves the slope formula:

• Identify the coordinates of the first point \(x_{1}\, y_{1}\) and the second point \(x_{2}\, y_{2}\).
• Apply the formula \(m = \frac{{y_{2} - y_{1}}}{{x_{2} - x_{1}}}\) to find the slope \(m\).

This formula gives us a ratio of the vertical change (rise) to the horizontal change (run) between two points that lie on the line. Slope can indicate various line orientations:

• A positive slope suggests the line tilts upwards as you move from left to right.
• A negative slope means the line tilts downwards.
• A zero slope indicates a horizontal line.
• An undefined slope corresponds to a vertical line.

Understanding the slope allows for insights into the line's direction and behavior on a graph. In this exercise, both lines had slopes calculated to be \(3\), indicating they rise equally for every unit increased along the x-axis.

Point-Slope Formula

The point-slope formula is a convenient method in coordinate geometry to find the equation of a line when one point on the line and the slope are known. The formula is:

• \( y - y_{1} = m(x - x_{1}) \)

Where \((x_{1}, y_{1})\) is a known point on the line, and \(m\) is the slope. This formula can be used to:

• Write the equation of a line quickly when you know a point on the line and its slope.
• Determine the position of a line relative to a coordinate system.

In practical application, if you've calculated the slope of a line and have a specific point, you can express the entire line's equation using this formula. This facilitates direct comparison with other lines or conversion into other forms like slope-intercept or standard form by further manipulation. In the exercise, using the point-slope form might not have been needed directly, but understanding it supports deeper grasp of the line's parallel nature.

Coordinate Geometry

Coordinate geometry, or analytic geometry, blends algebra and geometry to study geometric figures using a coordinate system. This allows for a structured method to describe and analyze geometric properties:

• Combining the algebraic technique of slope with geometric interpretations.
• Using coordinates to precisely define points and lines.
• Facilitating the understanding of geometric relationships, such as parallelism.

In our exercise, the task was to determine parallelism between lines through points using their slopes. This required:

• Converting the problem into algebraic form through slope calculation.
• Analyzing and comparing slopes to interpret geometric relationships.

Coordinate geometry provides the systematic approach to establish that two lines are parallel if they share the same slope. By applying these principles, the parallel nature of the two lines in the exercise is thoroughly validated, demonstrating how coordinate geometry helps bridge geometric intuition with precise algebraic reasoning.