Problem 31 - Calculus with Concepts in Calculus

Question

Decide which pairs of lines are parallel, which are perpendicular, and which are neither. For any pair that is not parallel, find the point of intersection. \(2 x+3 y=-1\) and \(2 y+2=3(x-1)\)

Step-by-Step Solution

Answer

The lines are neither parallel nor perpendicular and intersect at (1, -1).

Step 1: Rewrite Equations in Slope-Intercept Form

The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope. First, solve each equation for \(y\).\For the first equation, \(2x + 3y = -1\), solve for \(y\):\\[3y = -2x - 1\] \[y = -\frac{2}{3}x - \frac{1}{3}\]\The slope is \(-\frac{2}{3}\).\For the second equation, simplify and solve \(2y + 2 = 3(x - 1)\):\\[2y + 2 = 3x - 3\] \[2y = 3x - 3 - 2\] \[2y = 3x - 5\] \[y = \frac{3}{2}x - \frac{5}{2}\]\The slope is \(\frac{3}{2}\).

Step 2: Compare the Slopes

To determine if the lines are parallel or perpendicular, compare their slopes.\Parallel lines have equal slopes, and perpendicular lines have slopes that are negative reciprocals.\The slopes are \(-\frac{2}{3}\) and \(\frac{3}{2}\). Since they are neither equal nor negative reciprocals of each other, the lines are neither parallel nor perpendicular.

Step 3: Find the Point of Intersection

Since the lines are neither parallel nor perpendicular, they intersect at a point. Calculate the intersection point by solving \(2x + 3y = -1\) and \(y = \frac{3}{2}x - \frac{5}{2}\) simultaneously.Substitute \(y\) from the second equation into the first equation:\[2x + 3\left(\frac{3}{2}x - \frac{5}{2}\right) = -1\]\Simplify:\[2x + \frac{9}{2}x - \frac{15}{2} = -1\]\Combine terms:\[\frac{4}{2}x + \frac{9}{2}x = \frac{2}{2}\] \[\frac{13}{2}x = \frac{13}{2}\] \[x = 1\]Now substitute \(x = 1\) back into \(y = \frac{3}{2}x - \frac{5}{2}\):\[y = \frac{3}{2}(1) - \frac{5}{2}\] \[y = \frac{3}{2} - \frac{5}{2}\] \[y = -1\]The point of intersection is \((1, -1)\).

Key Concepts

Understanding Parallel Lines

Parallel lines are lines in a plane that remain the same distance apart and will never meet, no matter how far they are extended. This concept is central in analytical geometry as it helps in understanding the relationship between lines and their slopes. Here’s how you can easily identify parallel lines:

• They have the exact same slope. In mathematical language, if two lines have slopes \( m_1 \) and \( m_2 \), they are parallel if \( m_1 = m_2 \).
• They do not have to have the same intercept. That means parallel lines can cross different points on the y-axis.

When addressing equations, rewrite them into the slope-intercept form \( y = mx + b \) to easily compare slopes. In the given exercise, the slopes are \(-\frac{2}{3}\) and \(\frac{3}{2}\). Since these values are different, the lines are not parallel.

Characteristics of Perpendicular Lines

Perpendicular lines intersect at a right angle (90 degrees) and their slopes have a special relationship. In analytical geometry, this relationship is crucial for identifying when two lines are perpendicular:

• Their slopes are negative reciprocals of each other. This means if a line has a slope \( m_1 \), a line perpendicular to it will have a slope \( m_2 = -\frac{1}{m_1} \).

This property makes it easier to calculate and verify the perpendicular nature of lines without needing plotting. As seen in the solution, the slopes are \(-\frac{2}{3}\) and \(\frac{3}{2}\). Checking if they fit the negative reciprocal criteria (i.e., their product should be -1), we find that they do not fulfill this condition. Hence, these lines are not perpendicular.

Finding the Point of Intersection

When two lines are neither parallel nor perpendicular, they inevitably intersect at a certain point. Identifying this point is a key task in analytical geometry.To find the intersection point, we can solve the system of equations representing the lines by using substitution or elimination. Here's a simplified breakdown to understand this:

• Substitute one line's equation into the other if they are in the form \( y = ... \).
• For the given problem, the intersection involves substituting \( y = \frac{3}{2}x - \frac{5}{2} \) into \( 2x + 3y = -1 \).
• By solving, we find that \( x = 1 \) and then substituting back, \( y = -1 \).
• Therefore, the lines intersect at the point \((1, -1)\).

Recognizing the point of intersection helps us understand geometric relationships and is often used in various applications like navigation and plotting graphs.