Finding Equations Of Lines: A Step-by-Step Guide

Hey everyone! Today, we're diving into the fascinating world of linear equations. Specifically, we're going to learn how to determine the equation of a line based on different conditions. This is super useful, whether you're a math whiz or just trying to brush up on your skills. We'll be tackling four different scenarios, each with its own unique twist. So, grab your pencils, and let's get started!

1. Line Passing Through (-3, -2) and Parallel to Y = 2x + 4

Okay, guys, the first challenge is this: we need to find the equation of a line that goes through the point (-3, -2) and is parallel to the line Y = 2x + 4. Remember, parallel lines have the same slope. This is the key piece of information! Let's break this down step-by-step to make it crystal clear.

Firstly, we know that the given line, Y = 2x + 4, is in the slope-intercept form (y = mx + b). In this form, 'm' represents the slope, and 'b' is the y-intercept. In our case, the slope (m) of the given line is 2. Because our new line is parallel, it will also have a slope of 2. We can represent the equation of the new line as y = 2x + b, where 'b' is the y-intercept of the new line, which we have to find out. Now we need to figure out the value of 'b' using the fact that the line passes through the point (-3, -2). This means that when x = -3, y = -2. Let's substitute these values into our equation: -2 = 2*(-3) + b. Simplifying the equation, we get -2 = -6 + b. Adding 6 to both sides gives us b = 4. So, the y-intercept (b) of our new line is 4. Now that we have the slope (m = 2) and the y-intercept (b = 4), we can write the equation of the line as y = 2x + 4. Seems like the new line has the same formula as the one we compared. But wait, why? Because the lines are parallel. Therefore, the equation of the line that goes through the point (-3, -2) and is parallel to the line Y = 2x + 4 is, again, y = 2x + 4. It's the same, and the result is pretty impressive, right?

This method is super useful and can be used on many linear equations. This is how you conquer the first problem. Remember, the core concept here is understanding that parallel lines share the same slope. When you are asked to deal with parallel lines, the first thing you should do is to look for the slope and then replace it in the new equation. Finding 'b' is just like solving a puzzle, using the x and y coordinate given in the problem. Then you will find the final result.

2. Line Passing Through (1, 3) and Perpendicular to 5x + 6Y - 10 = 0

Alright, let's switch gears and tackle our second problem. We are tasked with finding the equation of a line that passes through the point (1, 3) and is perpendicular to the line 5x + 6Y - 10 = 0. When lines are perpendicular, the product of their slopes is -1. This is another crucial piece of information. Let's see how this works! First, we need to find the slope of the given line, 5x + 6Y - 10 = 0. To do this, we'll rewrite the equation in slope-intercept form (y = mx + b). Let's isolate Y: 6Y = -5x + 10. Dividing everything by 6, we get Y = (-5/6)x + (10/6). So, the slope of the given line is -5/6. Now, since our new line is perpendicular to this one, we need to find the negative reciprocal of -5/6. The negative reciprocal of -5/6 is 6/5. Therefore, the slope (m) of our new line is 6/5. We can represent the equation of the new line as y = (6/5)x + b, where 'b' is the y-intercept, which we still need to figure out. Next, we use the fact that the line passes through the point (1, 3). Substitute x = 1 and y = 3 into the equation: 3 = (6/5)*1 + b. Simplifying, we get 3 = 6/5 + b. To solve for 'b', we subtract 6/5 from both sides: b = 3 - 6/5. This equals b = 9/5. So, the y-intercept of our new line is 9/5. Now we have everything we need! The slope is 6/5, and the y-intercept is 9/5. The equation of the line is y = (6/5)x + 9/5.

To summarize, when dealing with perpendicular lines, we need to find the negative reciprocal of the slope of the given line and use it as the slope of the new line. Then, as always, use the given point to solve for the y-intercept. Once again, it is important to remember what kind of lines you are dealing with. Using the right formula, the results are pretty amazing, don't you think?

3. Line Parallel to the X-axis and Passing Through (-5, 3)

Okay, let's keep the momentum going! Now, we're dealing with a line that is parallel to the x-axis and passes through the point (-5, 3). Lines parallel to the x-axis are horizontal lines, and they have a slope of 0. That makes this problem pretty straightforward. The equation of a horizontal line is always in the form y = c, where 'c' is a constant value representing the y-coordinate of every point on the line. Since our line passes through the point (-5, 3), the y-coordinate is 3. Therefore, the equation of the line is y = 3. That's it! It is as simple as that. There's not much math involved here, which makes this problem really quick to solve. Remember that a horizontal line always has a constant y-value.

When you see a problem like this, remember the rules: lines that are parallel to the x-axis are horizontal and have an equation of the form y = constant. The y-value of the point is the constant we need. With just a little bit of observation, you can solve this problem pretty fast.

4. Line Perpendicular to the Y-axis and Passing Through (6, -5)

Last but not least, let's tackle the final scenario: finding the equation of a line that is perpendicular to the y-axis and passes through the point (6, -5). Lines perpendicular to the y-axis are horizontal lines, just like in our previous example. This means they also have a slope of 0. Therefore, the equation of this line is again in the form y = c, where 'c' is a constant. Since the line passes through the point (6, -5), the y-coordinate is -5. Hence, the equation of the line is y = -5. And that's all, folks! This problem is as easy as the previous one. The equation is incredibly simple to find.

Again, if you see a line that is perpendicular to the y-axis, you will be facing a horizontal line with an equation in the form y = constant. The y-coordinate is the constant, which means that finding the answer to this problem is easy. Always remember the relationship between the line and the axis. This will help you find the result.

Conclusion

And that's a wrap, guys! We've successfully navigated through four different types of line equations, and I hope you found it insightful. Remember, practice makes perfect. Keep working on these problems, and you'll become a pro in no time! So, keep up the good work and keep learning. If you have any questions, feel free to ask. Cheers!