Perpendicular Line Equation: Y = 3x - 2, Point (-1, -2)

Alright, let's dive into finding the equation of a line that's perpendicular to a given line and passes through a specific point. This is a classic problem in coordinate geometry, and we'll break it down step by step to make sure we get it right. We're looking for a line that's not just any line, but one that has a very particular relationship to another line and a definite location.

Understanding Perpendicular Lines

Before we jump into the math, let's get a solid grip on what perpendicular lines are all about. In simple terms, two lines are perpendicular if they intersect at a right angle—that's a 90-degree angle, for those keeping score at home. The crucial property that links perpendicular lines is their slopes. If you have one line with a slope of 'm', a line perpendicular to it will have a slope of '-1/m'. This is because the product of the slopes of two perpendicular lines is always -1. This inverse relationship is key to solving these types of problems.

Why is this important? Because the slope tells us how steeply the line rises or falls as we move from left to right. A positive slope means the line goes up, a negative slope means it goes down, and the larger the absolute value of the slope, the steeper the line. When lines are perpendicular, their slopes are negative reciprocals of each other, creating that perfect right angle at their intersection. So, when we're given a line and asked to find a perpendicular one, the first thing we need to do is flip and negate the slope!

Knowing this relationship makes finding the perpendicular line much easier. Instead of guessing and checking, we can use the slope of the given line to directly calculate the slope of the perpendicular line. This is an essential tool in our mathematical toolkit. With this understanding, we can tackle the problem with confidence.

Step-by-Step Solution

1. Identify the Slope of the Given Line

Our mission starts with the line y = 3x - 2. This equation is in slope-intercept form, which is y = mx + b, where 'm' is the slope and 'b' is the y-intercept. By comparing our given equation to the slope-intercept form, we can easily see that the slope of the given line is 3. So, m = 3. Remember, the slope is the number that's multiplied by x. In this case, it's nice and clear.

2. Calculate the Slope of the Perpendicular Line

Now that we know the slope of our given line is 3, we can find the slope of the line perpendicular to it. As we discussed earlier, the slope of a perpendicular line is the negative reciprocal of the original slope. This means we flip the fraction and change the sign. So, if the original slope is 3 (or 3/1), the perpendicular slope will be -1/3. Simple as that! This is a crucial step, so double-check your work to make sure you've got the negative reciprocal correct.

3. Use the Point-Slope Form

The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and 'm' is the slope. We know the slope of our perpendicular line is -1/3, and we know it passes through the point (-1, -2). Plug these values into the point-slope form: y - (-2) = (-1/3)(x - (-1)). This simplifies to y + 2 = (-1/3)(x + 1). The point-slope form is super handy because it lets us create the equation of a line using just a point and a slope.

4. Convert to Slope-Intercept Form (Optional)

While the point-slope form is perfectly valid, we often prefer to express the equation in slope-intercept form (y = mx + b) because it's easier to read and compare. To convert, distribute the -1/3 on the right side: y + 2 = (-1/3)x - 1/3. Then, subtract 2 from both sides to isolate y: y = (-1/3)x - 1/3 - 2. To combine the constants, we need a common denominator. Since 2 is the same as 6/3, we can rewrite the equation as y = (-1/3)x - 1/3 - 6/3. Finally, combine the fractions: y = (-1/3)x - 7/3. So, the equation of the line perpendicular to y = 3x - 2 and passing through (-1, -2) is y = (-1/3)x - 7/3.

5. Double-Check Your Work

Always double-check your work to make sure you didn't make any mistakes. Here are a few things to look for:

• Did you correctly find the negative reciprocal of the original slope?
• Did you correctly substitute the point (-1, -2) into the point-slope form?
• Did you correctly distribute and simplify the equation?
• Does your final equation seem reasonable?

If possible, graph both the original line and your perpendicular line to visually confirm that they intersect at a right angle and that your line passes through the given point. This visual check can often catch errors that you might miss when reviewing your calculations.

Common Mistakes to Avoid

When solving problems like this, it's easy to make common mistakes. Here are a few to watch out for:

• Forgetting to take the negative reciprocal: Remember, the slope of a perpendicular line is not just the reciprocal, it's the negative reciprocal. Don't forget to change the sign!
• Incorrectly distributing: Be careful when distributing the slope in the point-slope form. Make sure you multiply the slope by both terms inside the parentheses.
• Arithmetic errors: Simple arithmetic errors can throw off your entire solution. Double-check your calculations, especially when dealing with fractions.
• Plugging in the wrong point: Make sure you're using the correct coordinates for the given point. It's easy to mix up the x and y values.

Alternative Methods

While the method described above is the most common way to solve this type of problem, there are a couple of alternative approaches you could use.

Method 1: Using the General Form of a Line

Any line can be written in the general form Ax + By = C. If two lines, A1x + B1y = C1 and A2x + B2y = C2, are perpendicular, then A1A2 + B1B2 = 0. You can use this property to find the coefficients of the perpendicular line. First, rewrite the given equation y = 3x - 2 in general form: -3x + y = -2. So, A1 = -3 and B1 = 1. Now, let the perpendicular line be A2x + B2y = C2. We know that (-3)A2 + (1)B2 = 0, which means B2 = 3A2. Choose a convenient value for A2 (like 1), then B2 = 3. So, the equation of the perpendicular line is x + 3y = C2. Finally, plug in the point (-1, -2) to find C2: (-1) + 3(-2) = C2, which gives C2 = -7. Thus, the equation of the perpendicular line is x + 3y = -7. You can convert this to slope-intercept form to verify that it's the same as our previous answer.

Method 2: Visualizing the Solution

Sometimes, visualizing the problem can help you understand the solution better. Graph the given line and the point (-1, -2). Then, imagine a line passing through that point that is perpendicular to the given line. You can estimate the slope of the perpendicular line by visually inspecting the graph. This method is not as precise as the algebraic methods, but it can give you a good sense of whether your answer is reasonable.

Real-World Applications

Understanding perpendicular lines isn't just about solving math problems; it has practical applications in various fields. For example:

• Architecture: Architects use perpendicular lines to design buildings and ensure that walls are at right angles to the floor.
• Construction: Builders use perpendicular lines to lay foundations and ensure that structures are stable.
• Navigation: Navigators use perpendicular lines to determine direction and plot courses.
• Computer Graphics: Computer graphics designers use perpendicular lines to create realistic images and animations.

So, while it might seem like an abstract concept, understanding perpendicular lines is essential in many real-world scenarios.

Conclusion

Finding the equation of a line perpendicular to another line and passing through a given point might seem daunting at first, but by breaking it down into steps, it becomes manageable. Remember to find the negative reciprocal of the slope, use the point-slope form, and double-check your work. With practice, you'll be solving these problems like a pro. And who knows, maybe one day you'll use your knowledge of perpendicular lines to design a building, build a bridge, or navigate the high seas! Keep practicing and have fun with it! You've got this!