Problem 28 - 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. \(y=3-2 x\) and \(3 x+\frac{3}{2} y-4=0\)

Step-by-Step Solution

Answer

The lines are parallel.

Step 1: Write Equations in Slope-Intercept Form

First, we need to convert both equations into slope-intercept form (i.e., \(y = mx + b\)) to identify the slopes.1. The first equation is already in slope-intercept form: \(y=3-2x\). Here, \(m_1 = -2\).2. For the second equation \(3x + \frac{3}{2}y - 4 = 0\), solve for \(y\). Rearrange to find \(y\): \[\frac{3}{2}y = -3x + 4\] \[y = -2x + \frac{8}{3}\] Here, the slope \(m_2 = -2\).

Step 2: Determine if Lines Are Parallel

Lines are parallel if they have the same slope and different y-intercepts. In both equations, the slope is \(-2\), so \(m_1 = m_2\).The y-intercepts are different: \(3\) in the first equation and \(\frac{8}{3}\) in the second equation, so the lines are parallel.

Key Concepts

Slope-Intercept Form

The Slope-Intercept form is a way of writing the equation of a line, which allows us to understand the slope and the y-intercept directly from the equation. The general format is given by \( y = mx + b \), where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

Let's break it down:

• **Slope**: The slope \( m \) tells you how steep the line is. For example, if \( m = -2 \), for every 1 unit you move to the right along the x-axis, you go down 2 units on the y-axis. Think of it like the 'rise over run'.
• **Y-Intercept**: The y-intercept \( b \) is the value of \( y \) when \( x = 0 \). It's a quick way to see where the line starts on the y-axis.

When an equation like \( y = 3 - 2x \) is presented, you can immediately know that the slope \( m = -2 \) and the y-intercept \( b = 3 \). This form makes it incredibly straightforward to analyze and draw lines on a graph.

Parallel Lines

Parallel lines are lines in the same plane that never intersect, no matter how far they are extended. They have the same slope but different y-intercepts. The concept of parallel lines is frequently encountered when dealing with linear equations in slope-intercept form.

• **Same Slope**: For lines to be parallel, their slopes must be identical. No exceptions. If two lines have the same slope, they will never meet, just like train tracks that run alongside each other.
• **Different Y-Intercepts**: Even if the slopes are the same, the y-intercepts need to differ. Different y-intercepts ensure that the lines are distinct but keep the same direction.

In the exercise, both equations \( y = 3 - 2x \) and \( y = -2x + \frac{8}{3} \) were rewritten and confirmed to each have a slope of -2. Their y-intercepts, 3 and \(\frac{8}{3}\), are not the same, affirming their parallelism.Can you imagine them running along together, forever side-by-side, never crossing?

Point of Intersection

The point of intersection is where two different lines on a graph cross each other. It’s a crucial concept when lines are not parallel, which wasn't the task in this exercise but is still a very important idea in geometry and algebra.

• **Finding the Intersect**: Normally, to find this point, you would set the two equations equal to each other because at the intersection, both equations would have the same x and y values.
• **Solving for x and y**: By equating and simplifying, you solve for \( x \). Plug this value into either original equation to find the corresponding \( y \).

Although in our exercise the lines were parallel, meaning they never actually intersect, understanding this process can help tremendously when dealing with non-parallel, intersecting lines. At the intersection, it seems like you can almost imagine a painted dot where they cross paths. It's a wonderful 'aha' moment for any budding mathematician.