Perpendicular Line Equation: Step-by-Step Solution

Hey guys! Today, let's dive into a common math problem: finding the equation of a line that's perpendicular to another line and passes through a specific point. It might sound a bit complex, but don't worry, we'll break it down step by step. So, let's get started and learn how to solve these problems like a pro!

Understanding Perpendicular Lines

Before we jump into the problem, let's quickly recap what it means for lines to be perpendicular. Perpendicular lines are lines that intersect at a right angle (90 degrees). A key concept here is the relationship between their slopes. If you have two perpendicular lines, the product of their slopes is always -1. This means if one line has a slope of 'm', the slope of a line perpendicular to it will be '-1/m'. This inverse relationship is crucial for solving our problem. Think of it like this: if one line is steep going upwards, the perpendicular line will be gently sloping downwards, and vice versa. Understanding this relationship between slopes is fundamental for tackling problems involving perpendicular lines. For instance, if you know a line has a slope of 2, you immediately know that any line perpendicular to it will have a slope of -1/2. This simple rule allows us to quickly determine the slope of the perpendicular line, which is the first major step in finding its equation. So, when you encounter a problem asking for a perpendicular line, remember this key concept about slopes – it's your starting point.

Problem Statement

Alright, let's look at the specific problem we're tackling. We need to find the equation of a line that: 1) passes through the point (-6, 3), and 2) is perpendicular to the line given by the equation x + 3y = 12. This problem combines two important concepts in coordinate geometry: points and slopes. We have a specific point that our new line must go through, and we have a condition about its slope – it must be perpendicular to another line. To solve this, we'll first need to figure out the slope of the given line (x + 3y = 12). Once we have that, we can use the perpendicular slope relationship we discussed earlier to find the slope of our target line. Then, with the slope and the point (-6, 3), we can use a point-slope form to write the equation of the line. This step-by-step approach will help us to break down the problem into manageable parts. Remember, the key is to first understand what the problem is asking and then identify the tools and concepts you need to use. In this case, we're using our knowledge of perpendicular slopes and point-slope form to find the equation of the line. So, let's move on to the next step and start solving!

Step 1: Find the Slope of the Given Line

The first thing we need to do is figure out the slope of the given line, which is x + 3y = 12. To easily find the slope, we need to rewrite this equation in slope-intercept form. Remember, the slope-intercept form of a linear equation is y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. So, let's rearrange our equation: x + 3y = 12. First, we'll subtract 'x' from both sides, giving us 3y = -x + 12. Next, we'll divide both sides by 3 to isolate 'y': y = (-1/3)x + 4. Now, we have the equation in slope-intercept form! By looking at the equation, we can clearly see that the slope ('m') of the given line is -1/3. This is a crucial piece of information because it allows us to find the slope of the perpendicular line. Remember, the slope-intercept form is a powerful tool for quickly identifying the slope and y-intercept of a line. By converting the equation into this form, we've made it easy to extract the slope, which is the first step in solving our problem. So, now that we have the slope of the given line, we can move on to the next step: finding the slope of the perpendicular line.

Step 2: Determine the Slope of the Perpendicular Line

Now that we know the slope of the given line is -1/3, we can find the slope of the line perpendicular to it. Remember our rule: the slopes of perpendicular lines are negative reciprocals of each other. This means we need to flip the fraction and change the sign. So, if the slope of the given line is -1/3, the slope of the perpendicular line will be the negative reciprocal of -1/3. To find the negative reciprocal, we first flip the fraction, which gives us 3/1 or simply 3. Then, we change the sign. Since our original slope was negative (-1/3), the negative reciprocal will be positive. Therefore, the slope of the line perpendicular to x + 3y = 12 is 3. This step is super important because it gives us the 'm' value we need for our new line's equation. We now know that our target line has a slope of 3 and passes through the point (-6, 3). With this information, we're well on our way to finding the equation of the line. So, let's move on to the next step and use this information to write the equation of the line.

Step 3: Use the Point-Slope Form

Okay, we're getting closer! We now know the slope of our perpendicular line (m = 3) and a point it passes through (-6, 3). To write the equation of the line, we'll use the point-slope form. The point-slope form is a handy way to express the equation of a line when you know a point on the line (x₁, y₁) and its slope 'm'. The formula for the point-slope form is: y - y₁ = m(x - x₁). Let's plug in the values we know. Our point is (-6, 3), so x₁ = -6 and y₁ = 3. Our slope is m = 3. Substituting these values into the point-slope form, we get: y - 3 = 3(x - (-6)). Notice how we're carefully substituting the values into the correct places in the formula. The point-slope form is a powerful tool because it allows us to directly use the information we have (a point and a slope) to write the equation of the line. It's especially useful in situations like this, where we're given a point and a condition about the slope (being perpendicular to another line). So, now that we have the equation in point-slope form, we can simplify it further to get it into a more familiar form, like slope-intercept form. Let's move on to the final step and simplify our equation.

Step 4: Simplify to Slope-Intercept Form (Optional)

We have our equation in point-slope form: y - 3 = 3(x + 6). Now, let's simplify it to the slope-intercept form (y = mx + b), which is often considered the standard form for linear equations. First, we'll distribute the 3 on the right side of the equation: y - 3 = 3x + 18. Next, we'll add 3 to both sides to isolate 'y': y = 3x + 18 + 3. Finally, we combine the constants: y = 3x + 21. And there you have it! We've successfully converted our equation into slope-intercept form. The equation of the line that passes through the point (-6, 3) and is perpendicular to the line x + 3y = 12 is y = 3x + 21. This final form makes it easy to see the slope (3) and the y-intercept (21) of our line. Converting to slope-intercept form is an optional step, but it often makes the equation easier to interpret and use for graphing or other purposes. So, we've completed all the steps and found the equation of the line. Great job!

Final Answer

So, the general equation of the line that passes through the point (-6, 3) and is perpendicular to the line x + 3y = 12 is: y = 3x + 21. We did it! We started by understanding the concept of perpendicular lines and their slopes, then we worked step-by-step to find the equation of the line. We first found the slope of the given line, then used that to determine the slope of the perpendicular line. After that, we used the point-slope form to write the equation and finally simplified it to slope-intercept form. This problem demonstrates how breaking down a complex problem into smaller, manageable steps can make it much easier to solve. Remember to always identify the key concepts and tools you need, and then work through the problem systematically. You guys nailed it! Keep practicing, and you'll become a pro at solving these types of problems. Now you have a solid understanding of how to find the equation of a perpendicular line. Keep up the great work!