Problem 35 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the lines through each pair of points are parallel, perpendicular, or neither. $$(-5,-1)\( and \)(0,2) ;(-6,9)\( and \)(3,-6)$$

Step-by-Step Solution

Answer

The lines passing through the points (-5,-1) and (0,2), and (-6,9) and (3,-6) are perpendicular.

Step 1 - Determine the Slope of the First Line

The slope of a line passing through two points \((x1, y1)\) and \((x2, y2)\) is given by \( m1 = (y2 - y1) / (x2 - x1) \). Substituting the coordinates of the first pair of points into this equation, we get \( m1 = (2 - -1) / (0 - -5) = 3 / 5 \). This is the slope of the first line.

Step 2 - Determine the Slope of the Second Line

We use the same slope formula for the second line. Upon plugging the coordinates of the second pair of points into the equation, we find \( m2 = (-6 - 9) / (3 - -6) = -15 / 9 = -5/3 \). This is the slope of the second line.

Step 3 - Identify if the Lines are Parallel, Perpendicular, or Neither

From Step 1 and 2, we know that the slope of line 1, m1 = 3/5, and the slope of line 2, m2 = -5/3. If these lines were parallel, the slopes would be equal, but they aren't. If these lines were perpendicular, then the product of their slopes would be -1. Checking this, \( m1 * m2 = (3/5) * (-5/3) = -1 \), which is true. Therefore, the lines are perpendicular.

Key Concepts

Slope Calculations

Understanding slope calculations is essential when studying coordinate geometry. The slope of a line essentially describes its steepness and direction. Mathematically, it's represented as the rate of change between the y-coordinates and the x-coordinates of two distinct points on a line. To calculate it, you can use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
where \(m\) is the slope, and \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of two points on the line. In our exercise, by applying the formula, we've found the slopes of two lines by using pairs of points. One thing to remember is that if this formula results in a positive number, the line slopes upward from left to right, and if it's negative, it slopes downwards. When comparing two lines, if their slopes are equal, the lines are parallel. If the product of their slopes is -1, the lines are perpendicular.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a powerful branch of mathematics that combines algebra and geometry to describe the position of points, lines, and shapes. One of the most fundamental concepts in coordinate geometry is the use of the Cartesian coordinate system where each point is designated by a pair of numerical coordinates, which are the distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

When it comes to lines, understanding slopes and interpreting their relationships are key. For example, in the exercise given, once we've calculated the slopes of two lines using their points, we can compare these slopes to determine if the lines are parallel, perpendicular, or neither. In a graph, parallel lines never meet (they have the same slope), perpendicular lines intersect at a right angle, and if neither, the lines intersect but not at a right angle.

Linear Equations

Linear equations form the basis of lines in coordinate geometry. These equations can always be expressed in the form \[ y = mx + b \] where \(m\) represents the slope and \(b\) represents the y-intercept, the point where the line crosses the y-axis. It's fascinating how these simple equations can describe infinite points that form a straight line in a plane.

During the exercise, we indirectly used the concept of linear equations. Calculating the slopes of the lines can help us deduce the 'm' value in the linear equation. If you're tasked to check the orientation of lines, knowing their slopes (derived from the linear equations) can readily tell you if they are parallel or perpendicular. Remember, the slope is the key to understanding the relationship between lines – knowing how to calculate and interpret it can unlock a deeper comprehension of geometry and algebra.