Equation Mastery: Line Through Origin & Perpendicular

Hey everyone! Today, we're diving into a cool geometry problem: finding the equation of a line that does two things. First, it has to go right through the origin (that's the point (0,0)), and second, it needs to be perpendicular to another line. The given line's equation is x + 2y = 19, and we'll also be looking at 3x + 4y = 18, just for fun! This isn't just some boring math exercise, guys; it's a fundamental concept that pops up everywhere in science, engineering, and even computer graphics. Understanding how to find equations of perpendicular lines is super useful. So, let's break this down step-by-step and make it easy to understand.

Understanding Perpendicular Lines

Alright, before we jump into the problem, let's get some basics down. What exactly does "perpendicular" mean? In simple terms, it means the lines intersect (cross) at a right angle (90 degrees). Think of the corner of a square or a rectangle; that's a right angle. Now, a super important fact about perpendicular lines is their slopes are negative reciprocals of each other. What does that mean? Let's say one line has a slope of 2. The slope of a line perpendicular to it would be -1/2 (flip the fraction and change the sign). If the slope of a line is -3/4, the perpendicular slope would be 4/3. Got it? Cool! This relationship between slopes is the key to solving our problem. The slope is a measure of how steep a line is. A positive slope means the line goes up as you move to the right, a negative slope means it goes down, and a slope of zero means a flat, horizontal line. Perpendicular lines are like two roads that meet to form a perfect 'L' shape.

Now, why is this concept so important? Well, imagine you're designing a bridge. You need to make sure that the support beams are strong and stable. That often involves making sure that beams are perpendicular to each other to distribute the weight evenly. In computer graphics, understanding perpendicular lines is essential for creating realistic 3D models. Without this knowledge, the models would look flat and distorted. In science, understanding perpendicular lines is key to working with vectors and forces.

Finding the Slope of the Given Line(s)

Okay, let's get down to business. We need to figure out the slope of the line x + 2y = 19. The easiest way to do this is to rearrange the equation into slope-intercept form, which is y = mx + b, where 'm' is the slope and 'b' is the y-intercept. To do that, we first subtract 'x' from both sides of the equation, which gives us 2y = -x + 19. Then, we divide everything by 2 to isolate 'y', resulting in y = -1/2x + 19/2. So, the slope of the line x + 2y = 19 is -1/2. Simple, right? We can do the same for the line 3x + 4y = 18. Rearranging for y, we get 4y = -3x + 18. Dividing everything by 4 gets y = -3/4x + 18/4. So, the slope of the line 3x + 4y = 18 is -3/4.

So, the slope tells us how much the line goes up or down for every unit we move to the right. It gives us crucial information that allows us to compare lines and also find their relationship, whether they are parallel, intersecting, or perpendicular. The slope is important because it also gives us the angle that the line forms with the x-axis. In the case of the lines we're working with, they have different slopes, which means they are not parallel. If we compare them, we see that the line y = -1/2x + 19/2 and y = -3/4x + 18/4. The first one is more horizontal compared to the second one.

Determining the Slope of the Perpendicular Line

Now that we have the slope of the original line (-1/2), we can find the slope of the line perpendicular to it. Remember, the slopes of perpendicular lines are negative reciprocals of each other. So, we flip the fraction -1/2 to get 2/1 (or just 2), and then change the sign from negative to positive. Therefore, the slope of the line perpendicular to x + 2y = 19 is 2. For the second line, the slope is -3/4. The perpendicular slope will be 4/3. This is the slope of the line that we are trying to find the equation of. This means that for every 3 units you move to the right on the line, it goes up 4 units. With the slope, we now know the steepness of the perpendicular line. The next step is to put this information into an equation.

This concept is also important for understanding vector operations. In physics and engineering, vectors are used to represent quantities that have both magnitude and direction. Determining the perpendicularity of vectors and lines helps in determining concepts like work, torque, and moments. The slope gives the direction of the line in relation to the coordinate plane. Using the slope, you can find the angle between two lines. These can be useful in various calculations.

Finding the Equation of the Perpendicular Line Through the Origin

We're almost there, guys! We know the slope of the perpendicular line (2), and we know it passes through the origin (0,0). We can use the slope-intercept form (y = mx + b) again. We know 'm' (the slope) is 2. So, we can write the equation as y = 2x + b. To find 'b' (the y-intercept), we can plug in the coordinates of the origin (0,0). So, 0 = 2(0) + b. This simplifies to 0 = 0 + b, which means b = 0. This makes perfect sense because the line passes right through the origin, where the y-intercept is always 0. So, the equation of the line that is perpendicular to x + 2y = 19 and passes through the origin is y = 2x. Let's do the same thing for the equation 3x + 4y = 18, with the slope of 4/3 and going through the origin (0, 0). We have y = (4/3)x + b. When we plug in (0, 0), we have 0 = (4/3)0 + b. Thus, b is equal to 0. The equation of the line that is perpendicular to 3x + 4y = 18 and passes through the origin is y = (4/3)x.

So, what have we learned here? We know that, with the line through the origin, it means that b = 0. When we have an equation of the line given to us, like ax + by = c, all we have to do is get the slope (m) first. After we get the slope, then we can get the slope of the perpendicular line. When we know the slope of the perpendicular line, and it also goes through the origin, we can get the equation by the slope-intercept form. We will just put the slope into the m variable, and then we can say that the equation is y = mx. Also, it is important to know how to use this knowledge in real-world scenarios.

Conclusion

And there you have it! We've successfully found the equation of a line that is perpendicular to x + 2y = 19, also 3x + 4y = 18, and passes through the origin. This involved understanding the concept of perpendicular lines, figuring out slopes, and using the slope-intercept form. This problem is a great example of how different mathematical concepts work together. Keep practicing, and you'll become a pro at this in no time! Keep in mind that the concept of perpendicular lines is crucial in various fields and disciplines. It can be helpful in engineering design, computer graphics, or even architecture. This mathematical concept can solve real-world problems.

This is a basic problem, but it lays the foundation for understanding more complex geometrical problems. Perpendicular lines help in the visualization of many mathematical and real-world problems. Hope you enjoyed this lesson, and keep exploring the fascinating world of math! The next step is understanding the other equations of the line and using them to solve more complex problems. Then, it is time to learn other mathematical concepts.