Hey there, math enthusiasts! Today, we're diving into a fascinating problem involving points, lines, and slopes. We'll break down the question step by step and make sure you understand every twist and turn. So, buckle up and get ready to explore the world of coordinate geometry!
The Challenge: Finding the Slope
Okay, guys, let's get straight to the heart of the matter. Imagine we've got a point floating around in the coordinate plane. This point, helpfully named (p, q), has coordinates that are positive integers – meaning they're whole numbers greater than zero. Now, this point isn't just hanging out randomly; it's chilling on a line. This line has a special equation: y = mx + 3. The 'm' in this equation is what we call the slope, and that's precisely what we're trying to figure out. Our mission, should we choose to accept it, is to express this slope 'm' in terms of our friendly neighborhood p's and q's.
The question essentially asks: Given a point (p, q) that lies on the line y = mx + 3, where p and q are positive integers, how can we express the slope 'm' of the line using p and q?
This is a classic problem that combines the concepts of coordinate geometry and linear equations. To tackle this, we need to remember the fundamental relationship between a point on a line and the line's equation. If a point lies on a line, it means the coordinates of the point satisfy the equation of the line. This is our golden key to solving this puzzle. So, let's dive deeper into how we can use this key to unlock the solution!
The Power of the Line Equation
The heart of this problem lies in understanding what the equation of a line, y = mx + 3, really tells us. This equation is in what we call slope-intercept form. It's a super useful way to represent a line because it directly shows us two crucial pieces of information: the slope (m) and the y-intercept (the point where the line crosses the y-axis, which is 3 in this case).
The slope (m) is the rate at which the line is rising or falling. It tells us how much the y-value changes for every one unit change in the x-value. A positive slope means the line is going upwards as we move from left to right, while a negative slope means it's going downwards. A slope of zero means the line is horizontal. In our case, 'm' is the star of the show, and we're trying to find its value.
The y-intercept is the point where the line crosses the vertical y-axis. In the equation y = mx + 3, the y-intercept is 3. This means the line passes through the point (0, 3). Knowing the y-intercept gives us a fixed point on the line, which can be helpful in visualizing and understanding the line's position.
Now, the crucial connection here is that any point that lies on this line must have coordinates that make the equation true. This is where our point (p, q) comes into play. Since (p, q) lies on the line y = mx + 3, we know that when we substitute 'p' for 'x' and 'q' for 'y' in the equation, it must hold true. This gives us a powerful tool to work with, and we'll use it in the next section to isolate and find 'm'.
Substituting the Point (p, q)
This is where the magic happens! We know that the point (p, q) lies on the line y = mx + 3. This means that if we replace 'x' with 'p' and 'y' with 'q' in the equation, the equation will still be true. So, let's do it! Substituting, we get:
q = mp + 3
Now, look what we've got! We've transformed the original equation into a new equation that includes our variables 'p' and 'q', and most importantly, it still has 'm', the slope we're after. This is like finding the treasure chest – we're on the right track!
The goal now is to isolate 'm' on one side of the equation. We want to get 'm' all by itself so we can express it in terms of 'p' and 'q'. This is a classic algebraic manipulation, and it's surprisingly straightforward. Think of it like peeling an onion – we need to undo the operations that are attached to 'm' one by one until we get to the core.
The first step in isolating 'm' is to get rid of the '+ 3' on the right side of the equation. To do this, we can subtract 3 from both sides of the equation. Remember, whatever we do to one side of an equation, we must do to the other to keep it balanced. So, let's subtract 3 from both sides:
q - 3 = mp + 3 - 3
This simplifies to:
q - 3 = mp
We're getting closer! Now, 'm' is only being multiplied by 'p'. To get 'm' by itself, we need to undo this multiplication. What's the opposite of multiplication? Division! So, we'll divide both sides of the equation by 'p'.
Isolating the Slope 'm'
Alright, we're in the home stretch! We've got the equation q - 3 = mp, and our mission is to get 'm' all by itself. As we discussed, the key is to divide both sides of the equation by 'p'. This will undo the multiplication and leave us with 'm' on its own.
Dividing both sides by 'p', we get:
(q - 3) / p = mp / p
Now, the 'p' on the right side cancels out, leaving us with:
(q - 3) / p = m
And there you have it! We've successfully isolated 'm' and expressed it in terms of 'p' and 'q'. We've found the treasure!
This equation, m = (q - 3) / p, tells us exactly how the slope of the line is related to the coordinates of the point (p, q) that lies on it. The slope is simply the difference between the y-coordinate (q) and 3, all divided by the x-coordinate (p). This is a neat and elegant result that showcases the power of algebra and coordinate geometry.
But wait, there's one more important detail to consider. Remember that the problem stated that 'p' is a positive integer. This is crucial because we can't divide by zero. So, our expression for the slope, m = (q - 3) / p, is valid as long as p is not equal to zero. Since the problem explicitly tells us that p is a positive integer, we're all good! This is a good reminder to always pay attention to the given conditions in a problem, as they often hold important clues.
The Answer and Why It Matters
So, after all that algebraic maneuvering, we've arrived at the answer. The slope of the line, expressed in terms of p and q, is:
m = (q - 3) / p
Therefore, the correct answer is option A. (q-3)/p.
But let's take a moment to appreciate what we've accomplished here. We didn't just blindly apply a formula; we understood the underlying concepts and used them to solve the problem. This is the real beauty of mathematics – it's not just about memorizing rules, but about understanding the relationships between different ideas.
This problem highlights the connection between a line's equation, its slope, and the points that lie on it. It demonstrates how we can use algebraic manipulation to solve for unknown quantities. These are fundamental skills that are essential for success in mathematics and many other fields. Understanding these concepts will help you tackle more complex problems and see the elegance and interconnectedness of mathematical ideas.
Furthermore, the ability to express one variable in terms of others is a powerful tool in various areas of science, engineering, and economics. It allows us to model real-world phenomena and make predictions. So, the skills we've used in this problem are not just abstract mathematical concepts; they have practical applications in the world around us.
Wrapping Up
Great job, guys! We successfully navigated this problem and found the slope of the line in terms of p and q. We saw how the equation of a line and the coordinates of a point on the line are related. We used algebraic manipulation to isolate the slope and express it in the desired form.
Remember, the key to solving these types of problems is to break them down into smaller, manageable steps. Don't be afraid to use what you know and build upon it. And most importantly, practice, practice, practice! The more you work with these concepts, the more comfortable and confident you'll become.
Keep exploring the fascinating world of mathematics, and you'll be amazed at what you can discover!
