Problem 34 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the lines through each pair of points are parallel, perpendicular, or neither. $$(-1,-11)\( and \)(0,-5) ;(0,-8)\( and \)(12,-6)$$

Step-by-Step Solution

Answer

The lines are neither parallel nor perpendicular.

Step 1: Calculate the Slope of the First Line

The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Apply this formula using the points \( (-1, -11) \) and \( (0, -5) \).

Step 2: Calculate the Slope of the Second Line

Using the same slope formula, calculate the slope of the line passing through the points \( (0, -8) \) and \( (12, -6) \).

Step 3: Compare the Slopes

If the slopes are equal, the lines are parallel. If the product of the slopes is -1, the lines are perpendicular. Otherwise, the lines are classified as 'neither'.

Key Concepts

Calculating Slope

Understanding how to calculate the slope is essential for analyzing the relationship between two lines. The slope is a measure of the steepness or the inclination of a line. It's expressed as the ratio of the 'rise' (the vertical change) over the 'run' (the horizontal change) between two distinct points on the line.

To calculate the slope, we use two points located on the line, denoted as \( (x_1, y_1) \) and \( (x_2, y_2) \). The formula to find the slope \( m \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using this formula, one finds the difference between the y-coordinates, also known as 'rise', and divides it by the difference between the x-coordinates, known as 'run'. If you have points like \( (-1, -11) \) and \( (0, -5) \) from the exercise, plug them into your formula and perform the subtraction. This result will provide you with the slope of the line which can be used in further analysis.

Parallel and Perpendicular Lines

In geometry, analyzing the relationships of lines often comes down to identifying whether they are parallel, perpendicular, or neither. Parallel lines are distinct lines that never intersect, no matter how far they extend.

For two lines to be parallel, their slopes must be identical. The idea here is that if two lines are moving in the same direction and at the same angle, they'll always maintain the same distance between them - hence, they won't meet. On the other hand, perpendicular lines are those that intersect at a right angle. The mathematical condition for this is that the product of their slopes must equate to -1.

Therefore, when you have calculated the slopes of two lines as in our example exercise, you can determine the relationship by comparing their slopes: if they're equal, the lines are parallel; if the product of their slopes is -1, they intersect at a 90-degree angle, thus being perpendicular. If the lines meet neither condition, they're classified as 'neither'.

Slope Formula

The slope formula is the cornerstone for much of coordinate geometry. It's the key to unlocking the relationships between two points in a plane and consequently, determining the behavior of the line they form. The slope formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] may seem simple, but it's powerful. It provides us with a consistent way to quantify the direction and steepness of a line.

After finding the slopes for both lines from the exercise using this formula, you'll see that you are actually determining how much y increases or decreases, as x increases by one unit. Recognizing this will help in understanding and interpreting the slope as a rate of change. This formula is also the stepping-stone to further concepts such as equation of a line, graphing linear equations, and even calculus.