Finding The Equation Of A Line Given A Point And Slope
Hey guys! Let's dive into a super important concept in math: finding the equation of a line. Specifically, we're going to tackle how to write the equation of a line when you know a point it passes through and its slope. This is a fundamental skill in algebra and calculus, and it's really not as scary as it sounds. Trust me, once you get the hang of it, you'll be like, "Why was I ever worried about this?"
Understanding the Point-Slope Form
When dealing with the equation of a line, the point-slope form is your best friend in situations like this. This form is incredibly useful because it directly incorporates the information we're given: a point and a slope. So, what exactly is the point-slope form? It looks like this:
y - y₁ = m(x - x₁)
Where:
- x₁ and y₁ are the coordinates of the given point.
- m is the slope of the line.
- x and y are the variables that represent any point on the line.
The point-slope form is powerful because it allows us to construct the equation of a line directly from the information provided, without needing to first find the y-intercept. This is especially handy when the given point isn't the y-intercept, which is often the case in practical problems. Imagine you're plotting a course on a map, or designing a ramp – you'll frequently know a point your line needs to pass through and the steepness (slope) you need, but not necessarily where it crosses the y-axis. That's where this form shines.
To truly appreciate the beauty of the point-slope form, let's break it down. The term (y - y₁) represents the change in the y-coordinate relative to the given point, while (x - x₁) represents the change in the x-coordinate. The slope, m, as we know, is the ratio of the change in y to the change in x. So, the equation is essentially saying that for any other point (x, y) on the line, the ratio of the vertical change to the horizontal change is constant and equal to the slope. It's a concise way of capturing the fundamental property of a straight line: its constant rate of change.
Now, why is this so much better than trying to force everything into slope-intercept form (y = mx + b) right away? Think about it. If you were to use slope-intercept form directly, you'd first have to substitute the given point (x₁, y₁) and the slope m into the equation and then solve for b (the y-intercept). That's an extra step, and it increases the chance of making an algebraic error. The point-slope form cuts out that middleman, giving you a direct path from the given information to the equation of the line. It's like having a mathematical shortcut, and who doesn't love a good shortcut?
Applying the Point-Slope Form to Our Problem
Alright, let's get practical. Our problem gives us a point D(5, -2) and a slope m = 2/5. We want to find the standard form of the equation of the line passing through this point with this slope. The standard form of a linear equation is Ax + By = C, where A, B, and C are integers, and A is usually positive. We'll get to that in a bit, but first, let's plug our given information into the point-slope form. This is where the magic happens!
Our given point D(5, -2) tells us that x₁ = 5 and y₁ = -2. We also know that m = 2/5. Now we carefully substitute these values into our point-slope formula:
y - y₁ = m(x - x₁)
y - (-2) = (2/5)(x - 5)
See how we just replaced the symbols with the actual numbers? That's the key! Make sure you pay close attention to the signs, especially when you're dealing with negative numbers. A small mistake here can throw off the whole equation.
Now, let's simplify this equation step-by-step. First, we'll deal with that double negative on the left side:
y + 2 = (2/5)(x - 5)
Okay, we're making progress. The next step is to distribute the slope (2/5) across the terms inside the parentheses on the right side. Remember, distributing means multiplying the term outside the parentheses by each term inside:
y + 2 = (2/5)x - (2/5)(5)
y + 2 = (2/5)x - 2
Notice how we multiplied (2/5) by 5, which simplified to 2? Always look for opportunities to simplify as you go; it'll make the algebra much cleaner. We're getting closer to having the equation of our line in a more manageable form.
Converting to Standard Form
So, we've got our equation in point-slope form and we've simplified it a bit. But the question asks for the equation in standard form, which is Ax + By = C. This means we need to rearrange our equation so that the x and y terms are on the left side and the constant term is on the right side. We also want to get rid of any fractions and make sure the coefficient of x is positive. Buckle up, guys, we're in the home stretch!
Our current equation is:
y + 2 = (2/5)x - 2
The first thing we want to do is eliminate the fraction. We can do this by multiplying every term in the equation by the denominator, which is 5. This is a crucial step – you have to multiply everything to keep the equation balanced:
5(y + 2) = 5[(2/5)x - 2]
Now, distribute the 5 on both sides:
5y + 10 = 2x - 10
Look ma, no fractions! We're one step closer. Now we need to get the x and y terms on the same side and the constant terms on the other. Since we want the coefficient of x to be positive, let's move the y term to the right side and the constant term to the left side. We'll do this by subtracting 5y from both sides and adding 10 to both sides:
5y + 10 - 5y + 10 = 2x - 10 - 5y + 10
20 = 2x - 5y
Almost there! We just need to rewrite the equation with the x and y terms on the left side, which is purely a matter of convention:
2x - 5y = 20
And there you have it! This is the standard form equation of the line that passes through the point (5, -2) and has a slope of 2/5. We did it!
Summary
Let's recap what we've done. We started with a point and a slope, used the point-slope form to write the equation of the line, and then manipulated that equation into standard form. The key takeaways are:
- Point-slope form is your friend: y - y₁ = m(x - x₁)
- Substitute carefully: Pay close attention to signs.
- Simplify step-by-step: Distribute, combine like terms, and eliminate fractions.
- Standard form: Ax + By = C, where A, B, and C are integers and A is usually positive.
Finding the equation of a line might seem like a small thing, but it's a building block for so many other concepts in math and science. So, practice these steps, and you'll be well on your way to mastering linear equations. You've got this!
