Finding Perpendicular Line Equations: A Step-by-Step Guide

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Hey everyone! Today, we're diving into the world of linear equations, specifically focusing on how to find the equation of a line that's perpendicular to another. This is super useful in geometry, physics, and even in computer graphics! We'll break it down step-by-step, making sure it's easy to understand, even if you're just starting out. We're going to solve the problem of writing an equation in slope-intercept form for the line that passes through the point (−3,−2)(-3, -2) and is perpendicular to the equation y=−2x+4y = -2x + 4. Let's get started, shall we?

Understanding the Basics: Slope-Intercept Form and Perpendicular Lines

Alright, before we jump into the problem, let's make sure we're all on the same page. First off, what's slope-intercept form? It's simply a way of writing a linear equation as:

y = mx + b

Where:

  • m represents the slope of the line (how steep it is).
  • b represents the y-intercept (where the line crosses the y-axis).

Pretty straightforward, right? Now, the key concept here is perpendicular lines. These are lines that intersect each other at a 90-degree angle. The slopes of perpendicular lines have a special relationship: they are negative reciprocals of each other. This means if one line has a slope of m, the slope of a line perpendicular to it will be -1/m. For example, if a line has a slope of 2, a perpendicular line will have a slope of -1/2. If a line has a slope of -3/4, its perpendicular line will have a slope of 4/3. This concept is the heart of solving our problem.

So, why is this important? Because to find the equation of a perpendicular line, we need to know the slope of the original line. And we already know how to do that. The given equation y = -2x + 4 is already in slope-intercept form! We can see immediately that the slope of this line is -2. So, what's the slope of a line perpendicular to it? That's what we'll figure out next.

Now we understand the basic concepts, including slope-intercept form and perpendicular lines, we can now start solving the question by finding the slope of the perpendicular line and then the equation itself. Make sure that you understand the terms before continuing. This will make your understanding of the question easier. Let’s move to the next part of this guide.

Finding the Slope of the Perpendicular Line

Okay, let's find the slope of the line perpendicular to y = -2x + 4. As we just mentioned, the slope of the given line is -2. To find the slope of the perpendicular line, we need to take the negative reciprocal. So, here's how we do it:

  1. Start with the original slope: m = -2
  2. Flip the fraction: Since -2 can be written as -2/1, the reciprocal is -1/2.
  3. Change the sign: The negative of -1/2 is 1/2.

Therefore, the slope of the perpendicular line is 1/2. We'll call this m₁ = 1/2. Cool, huh? Now we know the slope of the line we're looking for. This is a crucial step! Remember, without the slope, we can't write the equation in slope-intercept form. Now that we have the slope, we can move on to the next step, which involves using a point on the line to find the y-intercept. Let’s go!

Remember, the slope of the perpendicular line is the negative reciprocal of the original line's slope. In other words, if the original line has a slope of -2, the perpendicular line has a slope of 1/2. We're getting closer to solving this problem!

Using the Point-Slope Form to Find the Equation

Alright, now we have the slope of our perpendicular line (1/2), and we also have a point that the line passes through: (-3, -2). With this information, we're ready to find the equation of the line. We will use what's called the point-slope form of a linear equation. The point-slope form is:

y - y₁ = m(x - x₁)

Where:

  • m is the slope.
  • (x₁, y₁) is a point on the line.

We know that m₁ = 1/2 and the point is (-3, -2). Let's plug these values into the point-slope form:

y - (-2) = (1/2)(x - (-3))

Now, let's simplify this:

y + 2 = (1/2)(x + 3)

From here, we're going to use the point-slope form. In this case, we have a slope of 1/2 and a point (-3, -2). Next, we will use the distributive property to simplify this equation further and get it closer to the slope-intercept form that we need for our final answer. Don’t worry; it's pretty simple and is the next step to finish solving this problem. Keep it up, guys!

Converting to Slope-Intercept Form

Almost there, guys! We've found the point-slope form of the equation. Now we just need to convert it into the slope-intercept form (y = mx + b). Here's how:

  1. Distribute the slope: Multiply (1/2) by both terms inside the parentheses:

    y + 2 = (1/2)x + (3/2)

  2. Isolate y: Subtract 2 from both sides of the equation. To do this, we need to convert 2 into a fraction with a denominator of 2, so that it's easy to subtract from 3/2.

    y = (1/2)x + (3/2) - 2 y = (1/2)x + (3/2) - (4/2)

  3. Simplify: Combine the constants:

    y = (1/2)x - (1/2)

And there you have it! The equation of the line that passes through the point (-3, -2) and is perpendicular to y = -2x + 4 is y = (1/2)x - (1/2). Congrats, you made it!

To convert the point-slope form into the slope-intercept form, we must first distribute the slope. Then, we need to isolate the y-variable. When simplifying, combine the constants and you should get the equation we're looking for! Congratulations on finishing this problem and this guide.

Conclusion: Putting It All Together

So, to recap, here's what we did:

  1. Found the slope of the original line.
  2. Calculated the negative reciprocal to find the slope of the perpendicular line.
  3. Used the point-slope form to write the equation.
  4. Converted the equation to slope-intercept form.

And we did it all to find the equation of a line that passes through a specific point and is perpendicular to a given line! This skill is incredibly useful in various areas of math and science. The key is understanding the relationship between the slopes of perpendicular lines and being comfortable with algebraic manipulation. If you practice, it will become second nature! Hopefully, this guide helped you with your understanding of this topic. Good luck on your mathematics journey, everyone! If you have any questions, feel free to ask!

Remember, practice makes perfect! Try working through similar problems on your own to solidify your understanding. You can change the point or the original equation to create new problems and then solve for the perpendicular equation. Feel free to come back to this guide if you need a refresher. You can also look for more guides online and practice to hone your understanding of this topic.