Find Perpendicular Line Equation: Step-by-Step Guide
Hey guys! Ever stumbled upon a math problem that looks like it's speaking another language? Well, today we're going to break down a super common one involving lines, slopes, and perpendicularity. We're tackling the kind of question you might see in algebra or geometry, and we'll make sure you understand each step along the way. So, let's dive into the challenge: Which equation in slope-intercept form represents a line that passes through the point (2,3) and is perpendicular to the line y-9=2/3(x+7)?
Understanding the Problem
Before we jump into calculations, let's make sure we grasp what the problem is asking. We need to find the equation of a line. But not just any line! This line has two very specific requirements:
- It must pass through the point (2, 3). This means if we plug in x = 2 and y = 3 into the equation of the line, it should hold true.
- It must be perpendicular to another line. The other line is given in a slightly disguised form: y - 9 = (2/3)(x + 7). We'll need to figure out what this line's slope is before we can find the slope of the perpendicular line.
Knowing these two conditions is key to solving the problem. We're essentially looking for a line that fits a specific mold, and we'll use our knowledge of slopes and equations to find it. The main keywords here are slope-intercept form, perpendicular, and point (2,3), so keep these in mind as we go through the solution.
Step 1: Unveiling the Slope of the Given Line
The first step in our quest is to figure out the slope of the line given by the equation y - 9 = (2/3)(x + 7). This equation isn't quite in the familiar slope-intercept form (y = mx + b), where 'm' represents the slope and 'b' represents the y-intercept. So, we need to do a little algebraic maneuvering to get it there. To transform the equation into slope-intercept form, we need to isolate 'y' on the left side of the equation. This involves distributing the (2/3) on the right side and then adding 9 to both sides. Let's walk through the process:
- Start with: y - 9 = (2/3)(x + 7)
- Distribute the (2/3): y - 9 = (2/3)x + (2/3) * 7 which simplifies to y - 9 = (2/3)x + 14/3
- Add 9 to both sides: y = (2/3)x + 14/3 + 9
Now, we need to combine the constants 14/3 and 9. To do this, we'll rewrite 9 as a fraction with a denominator of 3: 9 = 27/3. So the equation becomes:
- y = (2/3)x + 14/3 + 27/3
- y = (2/3)x + 41/3
Ta-da! We've successfully converted the equation into slope-intercept form: y = (2/3)x + 41/3. Now, it's super clear that the slope of this line is 2/3. Remember, the slope ('m') is the coefficient of 'x' when the equation is in slope-intercept form. Knowing this slope is crucial because it's the key to finding the slope of the line we're actually looking for – the one that's perpendicular to this one. The main focus here is on understanding how to convert equations and identify the slope, so make sure you're comfortable with this before moving on!
Step 2: Finding the Slope of the Perpendicular Line
Okay, we've nailed down the slope of the given line (it's 2/3). But remember, we're on a mission to find the equation of a line that's perpendicular to this one. So, what's the connection between the slopes of perpendicular lines? This is a crucial concept, guys!
Here's the magic rule: The slopes of perpendicular lines are negative reciprocals of each other.
What does that mean in plain English? It means we need to do two things to the slope of the given line (2/3) to find the slope of the perpendicular line:
- Flip it (find the reciprocal): The reciprocal of 2/3 is 3/2.
- Change the sign: Since 2/3 is positive, the negative reciprocal will be negative. So, we change 3/2 to -3/2.
Therefore, the slope of the line perpendicular to y = (2/3)x + 41/3 is -3/2. This is a critical step, so make sure you've got it! We've now got half the battle won – we know the slope of the line we're looking for. Now, we just need to use the other piece of information we were given: that the line passes through the point (2, 3). The key takeaway here is the relationship between slopes of perpendicular lines – memorize that negative reciprocal rule!
Step 3: Crafting the Equation Using Point-Slope Form
Alright, we're on the home stretch! We know the slope of our mystery line (-3/2) and a point it passes through (2, 3). Now, we need to weave these pieces of information together to create the equation of the line. And guess what? There's a perfect tool for this job: the point-slope form of a linear equation.
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
We've got all these ingredients! We know m = -3/2 and (x1, y1) = (2, 3). Let's plug them into the formula:
- y - 3 = (-3/2)(x - 2)
Boom! We've got the equation in point-slope form. But hold on, the problem asked for the equation in slope-intercept form (y = mx + b). So, we've got one more step to go: converting this equation.
To convert to slope-intercept form, we need to distribute the -3/2 on the right side and then isolate 'y' on the left side:
- y - 3 = (-3/2)x + (-3/2) * (-2)
- y - 3 = (-3/2)x + 3
- Add 3 to both sides: y = (-3/2)x + 3 + 3
- y = (-3/2)x + 6
And there you have it! We've successfully transformed the equation into slope-intercept form: y = (-3/2)x + 6. The point-slope form is a powerful tool, so make sure you understand how to use it. This step is all about putting the pieces together and using algebra to get to our final answer.
Step 4: Double-Checking Our Answer
We've arrived at an equation, but before we celebrate, let's make sure it's the real deal. How can we verify that y = (-3/2)x + 6 is indeed the equation of the line we're looking for? There are two things we can check:
- Does it pass through the point (2, 3)? To check this, we'll plug in x = 2 and y = 3 into the equation and see if it holds true:
- 3 = (-3/2)(2) + 6
- 3 = -3 + 6
- 3 = 3 🎉 It checks out!
- Is it perpendicular to the line y = (2/3)x + 41/3? We already made sure the slopes are negative reciprocals, so this should be good. But it's always a good idea to double-check! The slope of our line is -3/2, and the slope of the original line is 2/3. They are indeed negative reciprocals.
We've checked all the boxes! Our answer is solid. This verification step is super important – it's like proofreading your work before you turn it in. Always take the time to make sure your answer makes sense in the context of the problem.
The Final Answer
Phew! We made it through the maze of slopes, intercepts, and perpendicularity. The equation of the line that passes through the point (2, 3) and is perpendicular to the line y - 9 = (2/3)(x + 7) is:
y = (-3/2)x + 6
So, the correct answer is C. We've not only solved the problem, but we've also walked through the reasoning and steps involved. Hopefully, this has helped you understand the concepts better and feel more confident tackling similar problems in the future. Remember, math is like building with blocks – each concept builds on the previous one. Keep practicing, and you'll become a math whiz in no time! Congratulations, guys, you nailed it!
