Finding The Equation Of A Perpendicular Line Passing Through (4, 3)
Hey guys! Today, we're diving into a classic math problem: finding the equation of a line that passes through a specific point and is perpendicular to another line. Sounds a bit tricky? Don't worry; we'll break it down step by step. We're going to tackle the problem where we need to find a line that goes through the point (4, 3) and is perpendicular to the line defined by the equation y - 6 = -9(x + 8). We'll express our final answer in the ever-so-useful slope-intercept form (y = mx + b). So, grab your pencils, and let's get started!
Understanding the Basics: Slope-Intercept Form and Perpendicular Lines
Before we jump into solving the problem, let's refresh our understanding of a few key concepts. These concepts are the backbone of solving this type of problem, and having a firm grasp on them will make the entire process much smoother. We'll be using the slope-intercept form quite a bit, and the concept of perpendicular lines is central to the problem itself. So, let's make sure we're all on the same page. This foundational knowledge will not only help us solve this particular problem but also equip us to tackle similar challenges in the future.
Slope-Intercept Form: y = mx + b
The slope-intercept form is a way of writing linear equations that makes it super easy to identify the slope and y-intercept of a line. The equation looks like this: y = mx + b, where:
- m is the slope of the line. The slope tells us how steep the line is and whether it's going uphill or downhill. A positive slope means the line goes uphill from left to right, while a negative slope means it goes downhill.
- b is the y-intercept. The y-intercept is the point where the line crosses the y-axis. It's the value of y when x is equal to 0.
The slope-intercept form is incredibly useful because it allows us to quickly visualize the line represented by the equation. If we know the slope and the y-intercept, we can easily graph the line. Conversely, if we have the graph of a line, we can easily read off the slope and y-intercept and write the equation in slope-intercept form.
For example, if we have the equation y = 2x + 3, we know that the slope is 2 and the y-intercept is 3. This means the line goes uphill (positive slope) and crosses the y-axis at the point (0, 3).
Perpendicular Lines: Slopes and Their Relationship
Now, let's talk about perpendicular lines. Perpendicular lines are lines that intersect at a right angle (90 degrees). The slopes of perpendicular lines have a special relationship: they are negative reciprocals of each other.
What does that mean? If one line has a slope of m, then a line perpendicular to it will have a slope of -1/m. To find the negative reciprocal, you flip the fraction and change the sign. For example:
- If a line has a slope of 2 (which can be written as 2/1), the slope of a perpendicular line is -1/2.
- If a line has a slope of -3/4, the slope of a perpendicular line is 4/3.
- If a line has a slope of -9, the slope of a perpendicular line is 1/9.
Understanding this relationship is crucial for solving our problem. We're given a line, and we need to find a line perpendicular to it. This means we need to figure out the slope of the given line, and then find its negative reciprocal to get the slope of the line we're looking for.
Now that we've covered these basics, let's move on to the exciting part: actually solving the problem!
Step-by-Step Solution: Finding the Equation
Alright, let's dive into the actual problem-solving process. We have all the tools we need now: we understand the slope-intercept form, we know the relationship between the slopes of perpendicular lines, and we have our given information. Remember, we need to find the equation of a line that passes through the point (4, 3) and is perpendicular to the line defined by y - 6 = -9(x + 8). We'll take it one step at a time, so it's super clear. Think of it like building a house – we need a strong foundation before we can put up the walls and roof!
Step 1: Find the Slope of the Given Line
The first thing we need to do is find the slope of the line given by the equation y - 6 = -9(x + 8). To do this, we need to rewrite the equation in slope-intercept form (y = mx + b). This will allow us to easily identify the slope, which is the m value. This is a crucial first step because the slope of the line we're trying to find will be directly related to this slope.
Let's start by distributing the -9 on the right side of the equation:
y - 6 = -9x - 72
Next, we need to isolate y by adding 6 to both sides of the equation:
y = -9x - 72 + 6
Simplifying, we get:
y = -9x - 66
Now, the equation is in slope-intercept form! We can clearly see that the slope of the given line is -9. Remember, the slope is the coefficient of the x term. So, we've successfully found the slope of the line we were given. This is a big step forward! We're one step closer to finding the equation of the perpendicular line.
Step 2: Determine the Slope of the Perpendicular Line
Now that we know the slope of the given line is -9, we can find the slope of a line perpendicular to it. Remember, the slopes of perpendicular lines are negative reciprocals of each other. So, to find the slope of the perpendicular line, we need to flip the fraction and change the sign.
The slope of the given line is -9, which can be written as -9/1. Flipping the fraction gives us -1/9. Now, we change the sign from negative to positive, resulting in a slope of 1/9.
Therefore, the slope of the line perpendicular to y = -9x - 66 is 1/9. We've successfully found the slope of the line we're trying to find! This is a key piece of information, as it will be the m value in our slope-intercept form equation. We're making great progress!
Step 3: Use the Point-Slope Form to Find the Equation
We now know the slope of our perpendicular line (1/9) and a point it passes through (4, 3). This is perfect for using the point-slope form of a linear equation. The point-slope form is a handy tool for finding the equation of a line when you know a point on the line and its slope. The point-slope form looks like this:
y - y1 = m(x - x1)
where:
- m is the slope of the line
- (x1, y1) is a point on the line
In our case, m = 1/9 and (x1, y1) = (4, 3). Let's plug these values into the point-slope form:
y - 3 = (1/9)(x - 4)
This is the equation of our line in point-slope form. However, we want the equation in slope-intercept form (y = mx + b), so we need to do a little more work. But don't worry, we're almost there!
Step 4: Convert to Slope-Intercept Form
To convert our equation from point-slope form to slope-intercept form, we need to isolate y. Let's start by distributing the 1/9 on the right side of the equation:
y - 3 = (1/9)x - 4/9
Now, we add 3 to both sides of the equation to isolate y:
y = (1/9)x - 4/9 + 3
To add -4/9 and 3, we need to express 3 as a fraction with a denominator of 9. 3 is the same as 27/9, so we have:
y = (1/9)x - 4/9 + 27/9
Combining the fractions, we get:
y = (1/9)x + 23/9
And there we have it! Our equation is now in slope-intercept form. We've successfully converted from point-slope form to the familiar y = mx + b format.
Final Answer: The Equation of the Perpendicular Line
The equation of the line that passes through the point (4, 3) and is perpendicular to the line y - 6 = -9(x + 8) is:
y = (1/9)x + 23/9
This is our final answer! We've found the equation of the line that satisfies all the given conditions. It's in slope-intercept form, so we can easily see its slope (1/9) and y-intercept (23/9). We did it!
Wrapping Up: Key Takeaways
So, guys, we've successfully navigated this problem step by step. We started with the given information, broke down the problem into smaller, manageable parts, and used our knowledge of slope-intercept form and perpendicular lines to arrive at the solution. Let's recap the key takeaways from this exercise:
- Slope-intercept form (y = mx + b) is your friend. It makes it easy to identify the slope and y-intercept of a line.
- Perpendicular lines have slopes that are negative reciprocals of each other. This is a crucial relationship to remember.
- The point-slope form (y - y1 = m(x - x1)) is a powerful tool when you know a point on a line and its slope.
- Break down complex problems into smaller steps. This makes the problem less daunting and easier to solve.
By understanding these concepts and practicing these steps, you'll be well-equipped to tackle similar problems in the future. Keep practicing, and you'll become a pro at finding equations of lines! Remember, math can be fun, especially when you break it down and understand the underlying principles. Keep up the great work, and I'll catch you in the next math adventure!
