Problem 80 - College Algebra and Calculus: An Applied Approach

Question

Determine if the lines \(L_{1}\) and \(L_{2}\) passing through the indicated pairs of points are parallel, perpendicular, or neither. \(L_{1}:(3,6),(-6,0) ; L_{2}:(0,-1),\left(5, \frac{7}{3}\right)\)

Step-by-Step Solution

Answer

The lines \(L_{1}\) and \(L_{2}\) are parallel.

Step 1: Calculate the slope for line L1

The formula to find the slope (m) is \(m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\). For the line L1, the coordinates of two points are (3,6) and (-6,0). So, the slope of line L1 is calculated as \(m_{L1} = \frac{0 - 6}{-6 - 3} = \frac{6}{9} = \frac{2}{3}\).

Step 2: Calculate the slope for line L2

Similarly, for line L2, the coordinates of two points are (0,-1) and \( \left(5, \frac{7}{3}\right)\). So, the slope of line L2 is calculated as \(m_{L2} = \frac{\frac{7}{3} - (-1)}{5 - 0} = \frac{10}{15} = \frac{2}{3}\).

Step 3: Compare the slopes of L1 and L2

Now, compare the slopes of the two lines. If \(m_{L1} = m_{L2}\), the lines are parallel. If \(m_{L1} \cdot m_{L2} = -1\), the lines are perpendicular. If neither of these are true, the lines are neither parallel nor perpendicular. After substituting the values, we get \(m_{L1} = m_{L2} = \frac{2}{3}\). Hence, the lines are parallel.

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial in coordinate geometry. The slope, typically represented by the letter m, measures how steep a line is. It is defined as the ratio of the vertical change (\text{rise}) to the horizontal change (\text{run}) between two points on a line.To calculate the slope, use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For example, consider a line passing through points (3, 6) and (-6, 0). The slope of this line would be \( m = \frac{0 - 6}{-6 - 3} = \frac{-6}{-9} = \frac{2}{3} \). A positive slope like this indicates that the line rises as it goes from left to right, while a negative slope would indicate that the line falls.

Parallel Lines

Parallel lines are lines in a plane that never intersect, no matter how far they are extended. In coordinate geometry, two lines are parallel if and only if their slopes are equal. When examining the slope of parallel lines, recall that they have the same steepness and run in the same direction.In our example, both lines L1 and L2 have the slope of \( \frac{2}{3} \). This equality in slopes leads us to conclude that these are indeed parallel lines. It is important to note that even though the lines move in the same direction, they are not on top of each other—this is a common misconception.

Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle (90 degrees). In coordinate geometry, the criterion for two lines to be perpendicular is that the product of their slopes is -1. To represent this mathematically: if line L1 has slope m1, and line L2 has slope m2, then L1 is perpendicular to L2 if and only if m1 \( \cdot \) m2 = -1.If we take slopes m1 = \( \frac{2}{3} \) and m2 = \( -\frac{3}{2} \), we find that m1 \( \cdot \) m2 = \( \frac{2}{3} \) \( \cdot \) \( \frac{-3}{2} \) = -1. In this situation, the lines would be perpendicular. However, our example illustrates parallel lines and not perpendicular ones.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry by using a coordinate system. This system allows for geometrical problems to be solved algebraically and provides a way to describe geometrical shapes in numerical terms. Key concepts of coordinate geometry include points, lines, and the angles between them.With coordinates, we can precisely define the location of points, the equations of lines, and the relationship between lines such as being parallel or perpendicular. In the context of our exercise, by placing points on the coordinate plane and using the slope formula, we can analyze the relationship between lines, expanding our understanding of geometric principles through algebraic processes.