Finding The Real Value Of X For Collinear Points A(x, 6), B(-1, 0), And C(0, 2)

Hey guys! Today, we're diving into a cool math problem: figuring out when three points on a plane line up perfectly. This concept is called collinearity, and it's super useful in geometry and beyond. We're given three points: A(x, 6), B(-1, 0), and C(0, 2). Our mission? To find the real value of x that makes these points collinear. Sounds like a fun challenge, right? Let's break it down step by step!

Understanding Collinearity

Before we jump into the calculations, let's make sure we're all on the same page about what collinearity means. Collinear points are simply points that lie on the same straight line. Imagine drawing a line through these points – if they all fit snugly on that line, they're collinear. If one point is off the line, then they're not collinear. So, how do we mathematically check if points are collinear? There are a couple of ways, but one of the most common and straightforward methods involves using the concept of the determinant of a matrix formed by the coordinates of the points. This method is elegant because it directly relates to the area of the triangle formed by the points. If the points are collinear, the "triangle" collapses into a line, and its area becomes zero. This translates to the determinant of the matrix being zero.

Another intuitive way to think about collinearity is through the slopes of the lines formed by pairs of points. If points A, B, and C are collinear, the slope of line AB must be equal to the slope of line BC (and also the slope of line AC). This makes sense because if the slopes are different, the lines would intersect, and the points wouldn't lie on the same straight line. This slope-based approach is also very handy and can sometimes simplify the calculations, especially when dealing with simpler coordinates. However, the determinant method is more general and works well even when the coordinates are more complex or involve variables, like in our problem. The key takeaway here is that collinearity is a fundamental concept in geometry, and understanding it gives us powerful tools for solving various problems related to lines and points in a plane. Whether you're into coordinate geometry, linear algebra, or even computer graphics, grasping collinearity will definitely come in handy! So, let's keep this definition in mind as we move on to applying the determinant method to our specific problem.

The Determinant Method for Collinearity

Okay, so we know what collinearity means, now let's get into the nitty-gritty of how to check it using determinants. The determinant method is a slick way to determine if three points are collinear. Here's the idea: we form a matrix using the coordinates of the points, calculate its determinant, and if the determinant is zero, bingo! The points are collinear. If the determinant isn't zero, then they're not on the same line. The matrix we form looks like this:

| x₁  y₁  1 |
| x₂  y₂  1 |
| x₃  y₃  1 |

Where (x₁, y₁), (x₂, y₂), and (x₃, y₃) are the coordinates of our points. In our case, these are A(x, 6), B(-1, 0), and C(0, 2). So, we'll plug these coordinates into the matrix. Now, how do we calculate the determinant of this 3x3 matrix? There are a couple of ways to do it, but one common method is to use the rule of Sarrus. It involves multiplying the diagonals and adding or subtracting them in a specific pattern. Alternatively, you can use cofactor expansion, which is a more general method that works for matrices of any size. For a 3x3 matrix, the determinant calculation using the rule of Sarrus looks like this:

Determinant = (x₁ * y₂ * 1) + (y₁ * 1 * x₃) + (1 * x₂ * y₃) - (1 * y₂ * x₃) - (x₁ * 1 * y₃) - (y₁ * x₂ * 1)

It might look a bit intimidating at first, but once you get the hang of it, it's pretty straightforward. Remember, the key is to keep track of the signs and the order of multiplication. Now, why does this determinant thing work anyway? Well, the absolute value of the determinant is actually related to the area of the triangle formed by the three points. Specifically, the area of the triangle is half the absolute value of the determinant. So, if the points are collinear, they don't form a triangle (they form a line), and the area is zero. This means the determinant must be zero as well. This connection between the determinant and the area of a triangle gives us a powerful tool for checking collinearity. It's a beautiful example of how different areas of math, like linear algebra and geometry, come together to solve problems. So, with this method in our toolkit, we're ready to tackle our specific problem and find the value of x that makes our points A, B, and C collinear. Let's plug in the coordinates and see what we get!

Setting up the Determinant

Alright, guys, let's get our hands dirty with the numbers! We know the coordinates of our points: A(x, 6), B(-1, 0), and C(0, 2). We also know the determinant method for checking collinearity. So, the first thing we need to do is set up the determinant using these coordinates. Remember the matrix we talked about? It looks like this:

| x  6  1 |
| -1 0  1 |
| 0  2  1 |

See how we've plugged in the x and y coordinates of points A, B, and C into the first two columns, and the third column is just filled with 1s? That's our matrix! Now, to find out if these points are collinear, we need to calculate the determinant of this matrix and see if it equals zero. If it does, then the points are collinear. If it doesn't, then they're not. Simple as that! So, how do we calculate this determinant? We can use the rule of Sarrus, which we mentioned earlier. It's a handy trick for calculating the determinant of a 3x3 matrix. Let's quickly recap the rule of Sarrus. We multiply the diagonals of the matrix in a specific pattern. We add the products of the diagonals going from top-left to bottom-right, and we subtract the products of the diagonals going from top-right to bottom-left. It's a bit easier to see visually, but basically, you're multiplying three numbers at a time and either adding or subtracting the result. Now, let's apply this to our matrix. We'll multiply the diagonals, keep track of the signs, and then set the whole thing equal to zero (because we want to find the value of x that makes the points collinear, meaning the determinant must be zero). It might seem like a lot of steps, but don't worry, we'll take it one step at a time. The key is to be careful with the arithmetic and keep track of the signs. Once we've calculated the determinant, we'll have an equation in terms of x. Then, it's just a matter of solving for x. This is where our algebra skills come into play! So, let's get calculating. We've set up the determinant, we know the rule of Sarrus, now it's time to crank through the numbers and find out what value of x makes these points line up perfectly. Are you ready? Let's do it!

Calculating the Determinant and Solving for x

Okay, guys, time to put on our calculation hats! We have our matrix set up, and we're ready to calculate the determinant using the rule of Sarrus. Remember, we're aiming to find the value of x that makes the determinant equal to zero, which will tell us when the points are collinear. Let's walk through the calculation step by step. First, we multiply the diagonals going from top-left to bottom-right:

• x * 0 * 1 = 0
• 6 * 1 * 0 = 0
• 1 * -1 * 2 = -2

Now, we add these products together: 0 + 0 + (-2) = -2. Next, we multiply the diagonals going from top-right to bottom-left:

• 1 * 0 * 0 = 0
• x * 1 * 2 = 2x
• 6 * -1 * 1 = -6

We add these products together: 0 + 2x + (-6) = 2x - 6. Now, we subtract the second sum from the first sum: -2 - (2x - 6). Let's simplify this expression: -2 - 2x + 6 = -2x + 4. Remember, we want the determinant to be zero, so we set this expression equal to zero: -2x + 4 = 0. Now we have a simple equation to solve for x! Let's isolate x. First, subtract 4 from both sides: -2x = -4. Then, divide both sides by -2: x = 2. Woohoo! We've found the value of x that makes the points A, B, and C collinear. It's x = 2. That wasn't so bad, was it? We set up the determinant, carefully calculated it using the rule of Sarrus, and then solved the resulting equation. The key here is to be organized and pay attention to the signs. A little mistake in the arithmetic can throw off the whole answer. But we persevered, and we got it! Now, just to be extra sure, it's always a good idea to check our answer. We can plug x = 2 back into the original matrix and recalculate the determinant to make sure it's actually zero. Or, we can use another method for checking collinearity, like comparing the slopes of the lines formed by the points. This will give us even more confidence in our solution. So, let's take a moment to appreciate what we've accomplished. We've not only found the value of x, but we've also reinforced our understanding of collinearity and the determinant method. That's a pretty good math workout for one day! Let's move on to verifying our solution and making sure everything checks out.

Verifying the Solution

Awesome work, guys! We found that x = 2 makes the points A(x, 6), B(-1, 0), and C(0, 2) collinear. But, just like any good detectives, we need to verify our solution to be absolutely sure. There are a couple of ways we can do this. The first way is to plug x = 2 back into our determinant matrix and recalculate the determinant. If we get zero, that confirms our solution. The second way is to use the slope method. If the points are collinear, the slopes between any two pairs of points should be the same. Let's start with the determinant method verification. Our matrix with x = 2 looks like this:

| 2  6  1 |
| -1 0  1 |
| 0  2  1 |

Let's recalculate the determinant using the rule of Sarrus. The products of the diagonals from top-left to bottom-right are:

• 2 * 0 * 1 = 0
• 6 * 1 * 0 = 0
• 1 * -1 * 2 = -2

The sum is 0 + 0 + (-2) = -2. The products of the diagonals from top-right to bottom-left are:

• 1 * 0 * 0 = 0
• 2 * 1 * 2 = 4
• 6 * -1 * 1 = -6

The sum is 0 + 4 + (-6) = -2. Subtracting the second sum from the first sum: -2 - (-2) = 0. Bingo! The determinant is zero, which confirms that our value of x = 2 is correct. Now, let's try the slope method for even more confidence. The slope between two points (x₁, y₁) and (x₂, y₂) is given by the formula:

m = (y₂ - y₁) / (x₂ - x₁)

Let's calculate the slope between points B(-1, 0) and C(0, 2):

m_BC = (2 - 0) / (0 - (-1)) = 2 / 1 = 2

Now, let's calculate the slope between points A(2, 6) (remember, we found x = 2) and B(-1, 0):

m_AB = (6 - 0) / (2 - (-1)) = 6 / 3 = 2

The slopes are the same! This further confirms that our solution x = 2 is correct. We've successfully verified our answer using two different methods. That's how you know you've really nailed the problem! We not only found the value of x but also made sure it's the right one. This process of verification is super important in math and in life. It's always a good idea to double-check your work and make sure your answers make sense. So, let's take a moment to pat ourselves on the back for a job well done. We tackled a collinearity problem, used the determinant method, solved for x, and then verified our solution. That's a lot of math in one go! Let's wrap up our discussion with a final recap and some key takeaways.

Conclusion and Key Takeaways

Alright, mathletes! We've reached the end of our journey to find the real value of x that makes points A(x, 6), B(-1, 0), and C(0, 2) collinear. And guess what? We nailed it! We found that x = 2 is the magic number that makes these points line up perfectly. Let's take a quick recap of our adventure. First, we understood the concept of collinearity – points lying on the same straight line. Then, we learned about the determinant method, a powerful tool for checking collinearity using a matrix formed from the coordinates of the points. We set up the determinant using our given points, calculated it using the rule of Sarrus, and solved the resulting equation for x. We found x = 2. But we didn't stop there! We're not just about finding answers; we're about being sure of them. So, we verified our solution using two different methods: recalculating the determinant with x = 2 and comparing the slopes of the lines formed by the points. Both methods confirmed that our answer is correct. So, what are the key takeaways from this problem?

First, collinearity is a fundamental concept in geometry with practical applications in various fields. Understanding it gives you a powerful tool for solving problems involving lines and points. Second, the determinant method is an elegant and efficient way to check for collinearity. It connects linear algebra (determinants) with geometry (collinearity), showing how different areas of math are interconnected. Third, always verify your solutions! Whether it's plugging the value back into the original equation or using a different method, verification builds confidence and ensures accuracy. Finally, problem-solving in math is a journey. It involves understanding the concepts, applying the right tools, persevering through calculations, and verifying the results. It's not just about getting the answer; it's about the process of learning and growing your mathematical skills. So, keep exploring, keep questioning, and keep solving! Math is a fascinating world, and there's always something new to discover. And remember, practice makes perfect! The more problems you solve, the more comfortable you'll become with different concepts and techniques. So, keep challenging yourself, and you'll be amazed at what you can achieve. Until next time, happy mathing!