Finding The Equation Of A Perpendicular Line Through (5, 3)

Hey guys! Today, we're diving into a classic coordinate geometry problem: finding the equation of a line that's perpendicular to a given line and also passes through a specific point. In this case, we need to find the equation of a line perpendicular to the line 4x - 5y = 5 and passing through the point (5, 3). We'll break this down step-by-step, making sure it's super clear and easy to follow. So, let's put our math hats on and get started!

Understanding Perpendicular Lines

Before we jump into the calculations, let's quickly recap what it means for two lines to be perpendicular. Perpendicular lines are lines that intersect at a right angle (90 degrees). A crucial property of perpendicular lines is that their slopes are negative reciprocals of each other. This is the key concept we'll use to solve our problem. Imagine you have two lines; if one has a slope of 'm', the slope of a line perpendicular to it will be '-1/m'. This inverse and sign change is what makes the lines meet at a perfect 90-degree angle. For instance, if a line has a slope of 2, a line perpendicular to it will have a slope of -1/2. Remember this relationship, because it's the cornerstone of finding perpendicular lines. Now that we've refreshed our understanding of perpendicular lines, we're well-equipped to tackle the specific problem at hand. We're going to take the given line, figure out its slope, and then use the negative reciprocal to find the slope of our desired perpendicular line. This slope, combined with the given point, will allow us to write the equation of the line we're searching for. So, let's move on to the first step: finding the slope of the given line.

Step 1: Finding the Slope of the Given Line

The first line we're given is 4x - 5y = 5. To figure out the slope, we need to rewrite this equation in slope-intercept form, which is y = mx + b, where 'm' represents the slope and 'b' is the y-intercept. Let’s rearrange our equation step-by-step. Starting with 4x - 5y = 5, we want to isolate 'y' on one side. First, we subtract 4x from both sides, which gives us -5y = -4x + 5. Next, we divide both sides by -5 to solve for 'y'. This results in y = (4/5)x - 1. Ah-ha! Now we can clearly see the slope. The slope 'm' of the given line is 4/5. Remember, the slope tells us how steep the line is and in what direction it's heading. A positive slope, like ours, means the line is going upwards as you move from left to right. The value 4/5 tells us that for every 5 units we move to the right along the line, we move 4 units up. Now that we've successfully identified the slope of our given line, we're ready for the next crucial step: finding the slope of the line that's perpendicular to it. This is where the concept of negative reciprocals comes into play. We'll take the slope we just found and flip it, changing its sign to find the slope of our perpendicular line.

Step 2: Finding the Slope of the Perpendicular Line

Okay, so we know the slope of our original line is 4/5. Now, to find the slope of a line perpendicular to this, we need to take the negative reciprocal. Remember, that means flipping the fraction and changing the sign. So, if the original slope is 4/5, the negative reciprocal will be -5/4. This is the slope of our new line – the one we're trying to find the equation for! Think of it this way: the original line has a positive slope, going upwards, but our perpendicular line has a negative slope, going downwards. The steepness is also related, with the new slope being steeper in the opposite direction. This negative reciprocal relationship is what ensures that the two lines intersect at a perfect right angle. Now that we've got the slope of our perpendicular line, we're halfway there! We have one crucial piece of information. The next thing we need is a point that our line passes through. Luckily, the problem gives us that point: (5, 3). With a slope and a point, we can use the point-slope form of a line to build our equation. So, let's jump into that next step and put all the pieces together!

Step 3: Using the Point-Slope Form

We've got the slope of our perpendicular line, which is -5/4, and we know it passes through the point (5, 3). Now we're going to use the point-slope form of a linear equation to find the equation of our line. The point-slope form is a super handy formula: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line. In our case, m = -5/4, x₁ = 5, and y₁ = 3. Let’s plug these values into the formula: y - 3 = (-5/4)(x - 5). This is the equation of our line in point-slope form! But, we usually want our equation in slope-intercept form (y = mx + b) or standard form (Ax + By = C). So, the next step is to simplify and rearrange this equation. We'll start by distributing the -5/4 on the right side and then isolate 'y' to get it into slope-intercept form. This will give us a clear picture of the line's slope and y-intercept, making it easier to graph or compare with other lines. So, let's move on to the algebraic manipulation and get our equation into a more familiar form!

Step 4: Converting to Slope-Intercept Form

Alright, we've got our equation in point-slope form: y - 3 = (-5/4)(x - 5). Now let’s convert it to slope-intercept form (y = mx + b). First, we need to distribute the -5/4 on the right side of the equation. So, we multiply -5/4 by both x and -5: y - 3 = (-5/4)x + (25/4). Next, we want to isolate 'y' on the left side, so we add 3 to both sides of the equation. Remember, to add 3 to 25/4, we need to express 3 as a fraction with a denominator of 4, which is 12/4. So, we have: y = (-5/4)x + (25/4) + (12/4). Now we can add the fractions: y = (-5/4)x + (37/4). Ta-da! We've successfully converted our equation to slope-intercept form. We can clearly see that the slope is -5/4 (which we already knew) and the y-intercept is 37/4. This form is super useful for visualizing the line and understanding its behavior. However, sometimes we want our equation in standard form, which is Ax + By = C. So, let's take one more step and convert our equation to standard form.

Step 5: Converting to Standard Form (Optional)

We've got our equation in slope-intercept form: y = (-5/4)x + (37/4). To convert it to standard form (Ax + By = C), we need to get rid of the fractions and rearrange the terms. The first step is to eliminate the fractions by multiplying both sides of the equation by the denominator, which is 4: 4y = -5x + 37. Now, we want to get the x and y terms on the same side, so we add 5x to both sides: 5x + 4y = 37. Boom! We've got our equation in standard form. It looks nice and tidy, with integer coefficients and the constant term on the right side. While the slope-intercept form is great for identifying the slope and y-intercept, the standard form is often preferred for its clean appearance and sometimes makes it easier to work with in certain situations. Now, let's circle back to the original question and see which of the given options matches our solution.

Step 6: Checking the Answer

Okay, we've done all the hard work! We found the equation of the line perpendicular to 4x - 5y = 5 and passing through the point (5, 3). We converted it to both slope-intercept form (y = (-5/4)x + (37/4)) and standard form (5x + 4y = 37). Now, let's compare our answer with the given options to make sure we've nailed it. The options were:

• 4x - 5y = 5
• 5x + 4y = 37
• 4x + 5y = 5
• 5x - 4y = 8

Aha! Our equation in standard form, 5x + 4y = 37, exactly matches the second option. That means we've successfully found the correct answer. Guys, this is how we solve this type of problem. We first find the slope of the given line, then calculate the negative reciprocal to get the slope of the perpendicular line. We then use the point-slope form to create the equation and convert it to either slope-intercept or standard form, depending on what's needed. And finally, we double-check our answer against the given options. You guys did great! Now you're equipped to tackle similar problems with confidence.

So, there you have it, guys! Finding the equation of a line perpendicular to a given line and passing through a specific point might seem tricky at first, but by breaking it down into clear steps, it becomes totally manageable. We started by understanding the relationship between slopes of perpendicular lines, then we found the slope of our given line, calculated the negative reciprocal, used the point-slope form, converted to slope-intercept and standard forms, and finally, checked our answer. Remember, the key is to take it one step at a time and understand the concepts behind each step. With practice, these types of problems will become second nature. Keep up the awesome work, and remember, math can be fun when you approach it with a clear plan and a little bit of confidence!