Line Equation: Finding Slope, Intercept, And Point-Slope Form

by Andrew McMorgan 62 views

Hey guys! Let's dive into the fascinating world of linear equations. We're going to break down how to find the equation of a line when you're given a slope and a point it passes through. Specifically, we'll tackle a problem where the slope is 43\frac{4}{3} and the point is (3, 5). We’ll explore how to determine the correct statement about this line, focusing on the y-intercept, the slope-intercept equation, and the point-slope equation. This is super important for understanding linear relationships, and we'll make sure it's crystal clear by the end of this article. So, buckle up and get ready to master the art of line equations!

Understanding the Problem

Before we jump into solving, let's really get what the problem is asking. We've got a line, right? And we know two crucial things about it: its slope and a point it goes through. The slope, which is 43\frac{4}{3}, tells us how steep the line is and in what direction it's going. For every 3 units we move to the right on the graph (the run), we go up 4 units (the rise). Think of it like climbing a hill – a slope of 43\frac{4}{3} means you're going up pretty steeply!

The point (3, 5) is simply a specific location on the line. It tells us that when the x-coordinate is 3, the y-coordinate is 5. This is our anchor point, the spot where the line is definitely passing through. Now, our mission is to figure out the equation of this line. This equation is like the line's unique ID card – it tells us everything we need to know about it. We'll be looking at different forms of this equation, like the slope-intercept form (y = mx + b) and the point-slope form (y - y1 = m(x - x1)). Our main goal is to determine which statement about this particular line's equation is actually true. We need to figure out if the y-intercept is 4, if the slope-intercept equation is y = 43\frac{4}{3}x + 1, or if the point-slope equation matches our line. So, let’s get started and find out! Remember, understanding the core concepts is key to solving any math problem.

Finding the Equation of the Line

Okay, let's roll up our sleeves and find the equation of our line! We know the slope (m = 43\frac{4}{3}) and a point (3, 5). There are a couple of ways we can tackle this, but the point-slope form is often the easiest way to go when you have this information. Remember, the point-slope form looks like this: y - y1 = m(x - x1), where (x1, y1) is our known point. Let's plug in the values we know. Our point (3, 5) gives us x1 = 3 and y1 = 5, and our slope m is 43\frac{4}{3}. So, the equation becomes:

y - 5 = 43\frac{4}{3}(x - 3)

This is the point-slope form of the equation! Awesome, we've got one form down. Now, let's transform this into the slope-intercept form (y = mx + b), which will help us identify the y-intercept. To do that, we need to distribute the 43\frac{4}{3} on the right side and then isolate y. First, distribute the 43\frac{4}{3}:

y - 5 = 43\frac{4}{3}x - 43\frac{4}{3} * 3

Simplify the second term:

y - 5 = 43\frac{4}{3}x - 4

Now, add 5 to both sides to get y by itself:

y = 43\frac{4}{3}x - 4 + 5

Finally, simplify the constants:

y = 43\frac{4}{3}x + 1

Ta-da! We've got the slope-intercept form of the equation. This form is super useful because it directly tells us the slope (m) and the y-intercept (b). In our equation, y = 43\frac{4}{3}x + 1, the slope (m) is indeed 43\frac{4}{3}, and the y-intercept (b) is 1. Now, with both the point-slope and slope-intercept forms in hand, we’re ready to evaluate the given statements and see which one holds true. This process of converting between different forms of a linear equation is a powerful tool, so make sure you're comfortable with it!

Evaluating the Statements

Alright, we've successfully found the equation of the line in both point-slope and slope-intercept forms. Now comes the fun part: let's put on our detective hats and evaluate the statements to see which one is the real deal. We have three statements to consider:

A. The y-intercept is 4. B. The slope-intercept equation is y = 43\frac{4}{3}x + 1. C. The point-slope equation is something (we'll need to compare).

Let's start with the easiest one, statement A: "The y-intercept is 4." We've already found the slope-intercept form of the equation, which is y = 43\frac{4}{3}x + 1. Remember, in the slope-intercept form (y = mx + b), the 'b' represents the y-intercept. In our equation, b = 1, not 4. So, statement A is definitely false. We can cross that one off the list!

Next up is statement B: "The slope-intercept equation is y = 43\frac{4}{3}x + 1." Hey, that looks familiar! That's exactly what we calculated when we converted from the point-slope form. So, statement B is looking pretty good. It seems to be true, but let's not jump to conclusions just yet. We still need to check statement C.

Finally, let's think about statement C, which refers to the point-slope equation. We found the point-slope form to be y - 5 = 43\frac{4}{3}(x - 3). To properly evaluate statement C, we'd need to see the actual statement provided in the original problem. It's crucial to compare our derived point-slope equation with the one given in the options. However, based on our work so far, we know that our point-slope form, y - 5 = 43\frac{4}{3}(x - 3), is a valid representation of the line. Given that we've already confirmed statement B as true, and we've shown that statement A is false, statement B is most likely the correct answer, assuming statement C presents a different (and therefore incorrect) point-slope equation. Make sense? Always double-check, guys!

Conclusion

We did it! We successfully navigated the world of linear equations and pinpointed the correct statement. By understanding the slope-intercept and point-slope forms, and by carefully working through the algebra, we were able to determine that statement B, "The slope-intercept equation is y = 43\frac{4}{3}x + 1," is the true one. We also showed that the y-intercept is 1, not 4, making statement A false.

This whole process highlights the importance of understanding the different forms of a linear equation and how to convert between them. Remember, the point-slope form is super handy when you know a point and the slope, while the slope-intercept form is great for quickly identifying the slope and y-intercept. By mastering these concepts, you'll be well-equipped to tackle any linear equation problem that comes your way. Keep practicing, guys, and you'll become linear equation wizards in no time! This stuff is the building blocks for so many other areas of math and science, so it's really worth getting a solid handle on it. You got this!