Finding The Y-Intercept: Line Through (1,3) & (4,-3)

Hey guys! Today, we're diving into a classic math problem: finding the y-intercept of a line. Specifically, we're going to tackle a problem where we're given two points on the line and need to figure out that crucial 'b' value in the slope-intercept form, y = mx + b. This is a fundamental concept in algebra and geometry, and mastering it will definitely boost your math skills. So, let's break it down together, step by step, in a super friendly and easy-to-understand way.

Understanding the Basics: Slope-Intercept Form

Before we jump into the problem, let's quickly refresh our memory about the slope-intercept form of a linear equation. Remember that y = mx + b? This simple equation is the key to understanding straight lines on a graph.

• y: Represents the vertical coordinate on the graph.
• x: Represents the horizontal coordinate on the graph.
• m: This is the slope of the line. The slope tells us how steep the line is and whether it's going uphill (positive slope) or downhill (negative slope) as we move from left to right. It's essentially the "rise over run," or the change in y divided by the change in x.
• b: This is the y-intercept. It's the point where the line crosses the y-axis (the vertical axis). In other words, it's the value of y when x is equal to 0. Finding b is our main goal in this problem! This value is incredibly important because it anchors the line on the coordinate plane. Think of it as the starting point from which the line extends, dictated by its slope. A clear understanding of the y-intercept not only helps in plotting lines but also in interpreting linear relationships in real-world scenarios. For instance, if we're modeling the cost of a service with a linear equation, the y-intercept might represent the fixed initial fee, while the slope represents the variable cost per unit of service. Therefore, mastering the concept of the y-intercept is not just an academic exercise but a practical skill with broad applications.

The Problem: Finding 'b' Given Two Points

Okay, now let's get to the specific problem we're tackling. We're given two points, C(1, 3) and D(4, -3), and we know that a line passes through both of these points. Our mission, should we choose to accept it (and we do!), is to find the value of b, the y-intercept, when the equation of this line is written in slope-intercept form. This type of problem is common in algebra and geometry, and it's a great way to practice applying the concepts we just reviewed. The challenge here is that we don't have the equation of the line directly. We only have two points. But don't worry, that's more than enough information to solve this! We'll use these points to first find the slope of the line, and then we'll use the slope and one of the points to solve for b. It's like a mathematical treasure hunt, where each step gets us closer to our final answer. Remember, the key is to break the problem down into smaller, manageable steps. We're not trying to solve everything at once. We're taking it one piece at a time, making sure we understand each step before moving on to the next. This approach not only helps us solve the problem accurately but also reinforces our understanding of the underlying mathematical principles.

Step 1: Calculate the Slope (m)

The first thing we need to do is figure out the slope of the line. Remember, the slope (m) tells us how much the line rises or falls for every unit we move to the right. We can calculate the slope using the following formula:

• m = (y₂ - y₁) / (x₂ - x₁)

Where (x₁, y₁) and (x₂, y₂) are the coordinates of our two points. In our case, we have C(1, 3) and D(4, -3). So, let's plug in the values:

• x₁ = 1, y₁ = 3
• x₂ = 4, y₂ = -3

Now, let's substitute these values into the slope formula:

• m = (-3 - 3) / (4 - 1)
• m = -6 / 3
• m = -2

So, the slope of our line is -2. This means that for every 1 unit we move to the right along the line, we move 2 units down. A negative slope indicates that the line is decreasing as we move from left to right. Understanding the sign of the slope is crucial for visualizing the line and making sure our calculations make sense. A positive slope would mean the line is increasing, while a slope of 0 would indicate a horizontal line. Now that we have the slope, we're one step closer to finding the y-intercept. The slope is a critical piece of the puzzle, as it tells us the line's direction and steepness. With the slope in hand, we can now move on to the next step: using the slope and one of our points to solve for b.

Step 2: Use Slope-Intercept Form to Find 'b'

Now that we know the slope (m = -2), we can use the slope-intercept form (y = mx + b) and one of our points to solve for b. It doesn't matter which point we choose; we'll get the same answer either way. Let's use point C(1, 3). This means:

• x = 1
• y = 3

We already know m = -2, so let's plug these values into the slope-intercept equation:

• 3 = (-2)(1) + b

Now, let's simplify and solve for b:

• 3 = -2 + b
• 3 + 2 = b
• 5 = b

So, the y-intercept, b, is 5! That's it! We've found the value of b. It's important to remember that we could have used point D(4, -3) instead of point C(1, 3), and we would have arrived at the same answer. This is a good way to double-check your work and ensure that you haven't made any mistakes. The fact that the y-intercept is 5 tells us that the line crosses the y-axis at the point (0, 5). This is a significant piece of information for graphing the line or understanding its behavior. Now that we've found both the slope and the y-intercept, we have all the information we need to write the equation of the line in slope-intercept form. But for now, we've successfully answered the question: what is the value of b?

Step 3: (Optional) Verify with the Other Point

Just to be super sure (and because it's always a good practice to double-check your work!), let's use the other point, D(4, -3), to verify our answer. We'll plug in x = 4, y = -3, and m = -2 into the equation y = mx + b, and see if we still get b = 5.

• -3 = (-2)(4) + b
• -3 = -8 + b
• -3 + 8 = b
• 5 = b

Yay! It works! We got the same value for b, which confirms that our calculations are correct. This step is a fantastic way to build confidence in your solution and catch any potential errors. It's like having a built-in error-checking system. By verifying our answer with both points, we've essentially proven that the line with a slope of -2 and a y-intercept of 5 indeed passes through both points C(1, 3) and D(4, -3). This reinforces our understanding of the relationship between the slope, y-intercept, and points on a line. Now we can confidently say that we've not only found the y-intercept but also verified its accuracy.

Conclusion: The Y-Intercept is 5

So, there you have it, guys! We successfully found the y-intercept (b) of the line passing through points C(1, 3) and D(4, -3). The value of b is 5. We did this by first calculating the slope of the line and then using the slope-intercept form of the equation (y = mx + b) to solve for b. Remember, these steps are crucial for solving similar problems. We first found the slope, which told us the direction and steepness of the line. Then, we used the slope and one of the points to plug into the slope-intercept form and isolate b. Finally, we even verified our answer using the other point, just to be extra sure! This problem is a great example of how we can use basic algebraic concepts to solve geometric problems. And the skills we've used here – calculating slope, using the slope-intercept form, and verifying our answer – are fundamental tools in mathematics. So, make sure you practice these steps and feel confident in your ability to apply them to other problems. Keep up the awesome work, and remember, math can be fun and rewarding when we break it down into manageable steps!

I hope this explanation was helpful and easy to follow. Keep practicing, and you'll become a pro at finding y-intercepts in no time! You've got this! Remember, the key to mastering math is practice, practice, practice. So, try solving similar problems, and don't be afraid to ask for help when you need it. Happy calculating!