Parallel & Perpendicular Slopes: -6x - 8y = 9

Hey guys! Let's dive into some linear equations and figure out how to find the slopes of lines that are either parallel or perpendicular to a given line. In this case, we're focusing on the line expressed by the equation $-6x - 8y = 9$. Understanding these concepts is super useful in geometry and other areas of math, so let's break it down step by step.

Understanding Slope and Linear Equations

Before we jump into the specifics, let's make sure we're all on the same page about slope and how it relates to linear equations. The slope of a line tells us how steep it is and in what direction it's going. Mathematically, slope is defined as "rise over run," which means the change in the vertical direction (y-axis) divided by the change in the horizontal direction (x-axis). We often use the letter 'm' to represent slope.

Slope-Intercept Form

The most common way to represent a linear equation is in slope-intercept form: $y = mx + b$, where:

• 'y' is the dependent variable (usually plotted on the vertical axis)
• 'm' is the slope of the line
• 'x' is the independent variable (usually plotted on the horizontal axis)
• 'b' is the y-intercept (the point where the line crosses the y-axis)

This form is incredibly helpful because it immediately tells us the slope ('m') and the y-intercept ('b') of the line. To find the slopes of parallel and perpendicular lines, our first step will be to convert the given equation into this slope-intercept form. This involves isolating 'y' on one side of the equation, which will reveal the slope.

Parallel Lines

Parallel lines are lines that run in the same direction and never intersect. The key characteristic of parallel lines is that they have the same slope. Think of railroad tracks – they run parallel to each other, maintaining the same steepness and direction. So, if we find the slope of our given line, we automatically know the slope of any line parallel to it. This is a fundamental concept in coordinate geometry and is crucial for solving various problems related to linear relationships.

Perpendicular Lines

Perpendicular lines, on the other hand, intersect at a right angle (90 degrees). The relationship between their slopes is a bit different. If a line has a slope of 'm', then any line perpendicular to it will have a slope of '-1/m'. This is often referred to as the negative reciprocal of the original slope. To find the negative reciprocal, you flip the fraction (reciprocal) and change the sign (negative). Understanding this relationship is essential for constructing right angles and analyzing geometric shapes.

Converting the Equation to Slope-Intercept Form

Okay, now let's get to the nitty-gritty and convert our equation, $-6x - 8y = 9$, into slope-intercept form ($y = mx + b$). This will allow us to easily identify the slope of the given line. Remember, our goal is to isolate 'y' on one side of the equation. We'll do this by performing algebraic manipulations, ensuring we maintain the balance of the equation.

Here's how we do it:

1. Add 6x to both sides: This moves the '-6x' term to the right side of the equation.

   $-6x - 8y + 6x = 9 + 6x$

   $-8y = 6x + 9$

2. Divide both sides by -8: This isolates 'y' on the left side.

   $(-8y) / -8 = (6x + 9) / -8$

   y = -rac{6}{8}x - rac{9}{8}

3. Simplify the fraction: We can simplify the fraction -6/8 by dividing both the numerator and denominator by their greatest common divisor, which is 2.

   y = -rac{3}{4}x - rac{9}{8}

Now our equation is in slope-intercept form! We can clearly see that the slope (m) of the line is -3/4 and the y-intercept (b) is -9/8.

Finding the Slope of a Parallel Line

As we discussed earlier, parallel lines have the same slope. Since we've determined that the slope of the line $-6x - 8y = 9$ is -3/4, any line parallel to it will also have a slope of -3/4. That's it! Finding the slope of a parallel line is as straightforward as identifying the slope of the original line.

Think about it this way: If two lines have the same 'steepness' (slope), they'll run alongside each other without ever meeting. This fundamental property of parallel lines makes working with them much simpler.

Finding the Slope of a Perpendicular Line

Now let's tackle the slope of a perpendicular line. Remember, perpendicular lines intersect at a right angle, and their slopes are negative reciprocals of each other. This means we need to flip the fraction of our original slope and change its sign.

Our original slope is -3/4.

1. Flip the fraction (reciprocal): 4/3
2. Change the sign (negative reciprocal): Since our original slope was negative, the negative reciprocal will be positive. So, the slope becomes +4/3.

Therefore, the slope of any line perpendicular to $-6x - 8y = 9$ is 4/3. This seemingly simple transformation—taking the negative reciprocal—is a powerful tool in geometry and helps us understand the relationships between lines that meet at right angles.

Quick Review and Key Takeaways

Let's recap what we've learned:

• The slope of a line tells us its steepness and direction.
• The slope-intercept form of a linear equation is $y = mx + b$, where 'm' is the slope and 'b' is the y-intercept.
• Parallel lines have the same slope.
• Perpendicular lines have slopes that are negative reciprocals of each other.
• To find the slope of a parallel line, simply identify the slope of the original line.
• To find the slope of a perpendicular line, take the negative reciprocal of the original slope.

By converting the equation $-6x - 8y = 9$ to slope-intercept form, we found its slope to be -3/4. Therefore, the slope of any line parallel to it is also -3/4, and the slope of any line perpendicular to it is 4/3.

Why This Matters

Understanding parallel and perpendicular slopes isn't just a math exercise; it has practical applications in various fields. For example:

• Architecture and Engineering: Ensuring walls are perpendicular and floors are level often involves using these concepts.
• Navigation: Calculating directions and plotting courses rely on understanding angles and slopes.
• Computer Graphics: Creating realistic images and animations often uses linear equations and slope calculations.

So, mastering these concepts opens doors to a broader understanding of the world around us. Keep practicing, and you'll be a slope-finding pro in no time!

Practice Problems

To solidify your understanding, try these practice problems:

1. Find the slopes of lines parallel and perpendicular to the line $2x + 5y = 10$.
2. A line has a slope of 2. What is the slope of a line perpendicular to it?
3. Are the lines $y = 3x + 1$ and $y = 3x - 5$ parallel? Why or why not?

Work through these, and you'll be well on your way to mastering parallel and perpendicular slopes. Good luck, and happy calculating!