Perpendicular Line Equation: Slope-Intercept Form Guide

by Andrew McMorgan 56 views

Hey guys! Ever find yourself scratching your head over perpendicular lines and slope-intercept forms? Don't sweat it! We're going to break down a super common problem step-by-step, so you'll be a pro in no time. Let’s dive into finding the equation of a line that's perpendicular to a given line and passes through a specific point. Trust me; it's way easier than it sounds!

Understanding the Basics

Before we jump into the problem, let's quickly recap some essential concepts. The slope-intercept form of a line is given by the equation y=mx+b{y = mx + b}, where m{m} is the slope and b{b} is the y-intercept. This form is super useful because it tells us two crucial things about the line: how steep it is (the slope) and where it crosses the y-axis (the y-intercept).

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. If one line has a slope of m{m}, then a line perpendicular to it will have a slope of −1m{-\frac{1}{m}}. This is a key concept to remember!

Knowing these basics, we can proceed with the problem at hand. We need to find the equation of a line that not only is perpendicular to a given line but also passes through a specific point. This involves a few steps, but each one is manageable with a clear understanding of these foundational ideas. So, keep these concepts in mind as we move forward, and you'll find the whole process much smoother. Let’s keep rolling!

Step 1: Identify the Slope of the Given Line

Alright, the first thing we gotta do is figure out the slope of the line we're given: y−4=23(x−6){y - 4 = \frac{2}{3}(x - 6)}. To make things easier, let's rewrite this equation in slope-intercept form, which is y=mx+b{y = mx + b}. This will allow us to easily identify the slope.

Starting with y−4=23(x−6){y - 4 = \frac{2}{3}(x - 6)}, distribute the 23{\frac{2}{3}} on the right side:

y−4=23x−23⋅6{y - 4 = \frac{2}{3}x - \frac{2}{3} \cdot 6}

Simplify the equation:

y−4=23x−4{y - 4 = \frac{2}{3}x - 4}

Now, add 4 to both sides to isolate y{y}:

y=23x−4+4{y = \frac{2}{3}x - 4 + 4}

y=23x{y = \frac{2}{3}x}

Now that we have the equation in slope-intercept form, y=23x{y = \frac{2}{3}x}, we can easily see that the slope of the given line is 23{\frac{2}{3}}. This is a crucial piece of information because we need the slope of a line perpendicular to this one. Remember, the slopes of perpendicular lines are negative reciprocals. This means we're not just flipping the fraction; we're also changing the sign. The original slope is positive, so the perpendicular slope will be negative.

So, the slope of the line we're trying to find will be the negative reciprocal of 23{\frac{2}{3}}, which is −32{-\frac{3}{2}}. Got it? Great! Now, with this new slope in hand, we're ready to move on to the next step: using this slope and the given point to find the equation of our perpendicular line.

Step 2: Determine the Slope of the Perpendicular Line

Okay, now that we know the slope of the given line is 23{\frac{2}{3}}, let's find the slope of the line perpendicular to it. Remember, the slopes of perpendicular lines are negative reciprocals of each other. This means we flip the fraction and change the sign.

So, if the original slope is 23{\frac{2}{3}}, the perpendicular slope will be −32{-\frac{3}{2}}. This is a super important step because this new slope is what we'll use to build the equation of our new line. Make sure you're comfortable with this idea of negative reciprocals; it comes up all the time in these types of problems.

Think of it this way: if a line has a positive slope, a line perpendicular to it will have a negative slope, and vice versa. This makes sense visually, right? One line goes up as you move from left to right, and the other goes down. Also, flipping the fraction ensures that the lines meet at a perfect 90-degree angle. Without this negative reciprocal relationship, the lines wouldn't be perpendicular.

With the perpendicular slope −32{-\frac{3}{2}} in our toolbox, we can now move on to using the point-slope form to find the full equation of the line. We're getting closer! Keep up the great work, and you'll have this problem solved in no time!

Step 3: Use the Point-Slope Form

Now that we have the slope of the perpendicular line (−32{-\frac{3}{2}}) and a point it passes through ((−2,−2){(-2, -2)}), we can use the point-slope form to find the equation of the line. The point-slope form is given by:

y−y1=m(x−x1){y - y_1 = m(x - x_1)}

where (x1,y1){(x_1, y_1)} is the given point and m{m} is the slope. In our case, (x1,y1)=(−2,−2){(x_1, y_1) = (-2, -2)} and m=−32{m = -\frac{3}{2}}.

Plug these values into the point-slope form:

y−(−2)=−32(x−(−2)){y - (-2) = -\frac{3}{2}(x - (-2))}

Simplify the equation:

y+2=−32(x+2){y + 2 = -\frac{3}{2}(x + 2)}

This is the equation of the line in point-slope form. While this form is perfectly valid, we usually want the equation in slope-intercept form (y=mx+b{y = mx + b}). So, let's move on to the next step to convert it.

Using the point-slope form is super handy because it allows us to directly plug in the slope and the coordinates of a point on the line. This avoids having to solve for the y-intercept separately. Just remember the formula and plug in the values carefully, paying attention to the signs. Once you get comfortable with the point-slope form, you'll find it's a quick and efficient way to find the equation of a line when you know its slope and a point it passes through. Keep practicing, and it'll become second nature!

Step 4: Convert to Slope-Intercept Form

Okay, we've got our equation in point-slope form: y+2=−32(x+2){y + 2 = -\frac{3}{2}(x + 2)}. Now, let's convert it to slope-intercept form (y=mx+b{y = mx + b}) so it's easier to read and use. To do this, we need to distribute and then isolate y{y}.

First, distribute the −32{-\frac{3}{2}} on the right side:

y+2=−32x−32⋅2{y + 2 = -\frac{3}{2}x - \frac{3}{2} \cdot 2}

Simplify the equation:

y+2=−32x−3{y + 2 = -\frac{3}{2}x - 3}

Now, subtract 2 from both sides to isolate y{y}:

y=−32x−3−2{y = -\frac{3}{2}x - 3 - 2}

y=−32x−5{y = -\frac{3}{2}x - 5}

And there you have it! The equation of the line in slope-intercept form is y=−32x−5{y = -\frac{3}{2}x - 5}. This tells us that the line has a slope of −32{-\frac{3}{2}} and a y-intercept of -5.

Converting to slope-intercept form is a crucial skill because it allows us to quickly identify the slope and y-intercept of a line. This makes it easy to graph the line or compare it to other lines. So, make sure you're comfortable with distributing and isolating y{y} to get the equation in this form. With a little practice, you'll be converting equations in no time, and you'll be one step closer to mastering linear equations!

Conclusion

So, to recap, we found the equation of a line perpendicular to y−4=23(x−6){y - 4 = \frac{2}{3}(x - 6)} and passing through the point (−2,−2){(-2, -2)}. We first identified the slope of the given line, then found the negative reciprocal to get the slope of the perpendicular line. Next, we used the point-slope form to create an equation, and finally, we converted that equation to slope-intercept form.

The equation of the line is:

y=−32x−5{y = -\frac{3}{2}x - 5}

Understanding how to find equations of perpendicular lines is super useful in many areas of math and science. Whether you're working on geometry problems or analyzing data, this skill will definitely come in handy. So, keep practicing, and you'll become a pro at finding these equations in no time!

Great job, everyone! You've now tackled a challenging problem and come out on top. Keep up the awesome work, and you'll be mastering all sorts of math concepts in no time!