Finding 'a': A Point On A Line's Journey

Hey math enthusiasts! Let's dive into a classic geometry problem: finding the value of 'a'. We're given a line that gracefully glides through two points, and we know another point resides on this very line. Our mission? To pinpoint the exact location of this third point. It's like a treasure hunt, but instead of gold, we're seeking a specific coordinate. Ready to unravel this geometric puzzle? Let's get started!

Unveiling the Equation: The Slope-Intercept Form

First things first, we need a roadmap, and in the world of lines, that roadmap is the equation. Remember the good ol' slope-intercept form? It's our trusty guide here. The general form looks like this: y = mx + b. Where 'm' represents the slope (how steeply the line climbs or descends), and 'b' is the y-intercept (where the line kisses the y-axis). Our goal is to build this equation specific to the line that contains all the points.

To find the slope ('m'), we use the slope formula. It goes like this: m = (y₂ - y₁) / (x₂ - x₁). We have two points (-1, -5) and (4, 5). Let's call (-1, -5) point one (x₁, y₁) and (4, 5) point two (x₂, y₂). Plugging these values into the slope formula, we get: m = (5 - (-5)) / (4 - (-1)). Simplifying this, we get m = 10 / 5, which gives us m = 2. Awesome, the slope is 2! This means that for every 1 unit we move to the right on the x-axis, the line climbs 2 units on the y-axis. The line has an upward slant.

Now, how do we find 'b', the y-intercept? We can use the slope and one of the points we're given. Let's use the point (4, 5) and plug the values into our slope-intercept equation: y = mx + b. Which becomes: 5 = 2(4) + b. Solving this equation, we get: 5 = 8 + b, subtract 8 from both sides, and b = -3. Okay, we have all the components of our line equation. The equation of the line is y = 2x - 3.

Now we've successfully crafted the equation of the line. This is the equation that all the points on the line must satisfy. It's like the secret password to enter the line's exclusive club.

Putting it Together: The Equation of the Line

So, to recap, our equation is y = 2x - 3. Now it's time to test if the points belong on the line. We know that the original two points are on the line, but we can verify this to ensure that the equation is correct. Plugging in the point (-1, -5) into the equation we get: -5 = 2*(-1) - 3, -5 = -2 - 3, -5 = -5. The point lies on the line. Plugging in the point (4, 5) into the equation we get: 5 = 2*(4) - 3, 5 = 8 - 3, 5 = 5. The point lies on the line. Great, our equation is accurate and describes the line where all our points will exist.

The Quest for 'a': Finding the Missing Coordinate

We know that the point (a, 1) is also on this line. This means that if we substitute '1' for 'y' and 'a' for 'x' in our equation, the equation must hold true. It's like saying, "If this point is on the line, it must obey the rules of the line's equation." Thus, using our equation y = 2x - 3, we substitute the values: 1 = 2a - 3. We're now one step closer to our goal.

Our focus has shifted to isolating 'a'. To do this, we need to get 'a' alone on one side of the equation. First, add 3 to both sides of the equation: 1 + 3 = 2a - 3 + 3, which simplifies to 4 = 2a. Now, to solve for 'a', divide both sides of the equation by 2: 4/2 = 2a/2. Finally, we arrive at the answer: a = 2. And there you have it, folks! The value of 'a' is 2. The missing coordinate is (2, 1). We've successfully used the equation to find a missing value.

Checking our Work: Does It Make Sense?

Before we celebrate, let's take a moment to reflect. Does the answer make sense? Does the point (2, 1) look like it could be on the line? Recall our slope is 2. The y-intercept is -3. Let's check this point using the original points. Using the point (-1, -5) and (2, 1), we can find the slope. m = (1 - (-5)) / (2 - (-1)), m = 6 / 3, m = 2. It matches! Using the point (4, 5) and (2, 1), we can find the slope. m = (5 - 1) / (4 - 2), m = 4 / 2, m = 2. It also matches! It does. It fits within the pattern we established. It's a key part of the line. So, it appears we've conquered the problem and solved for 'a'. It's always a good idea to confirm your solutions. This step ensures that we have the right answer.

Conclusion: A Line's Journey Completed

And there you have it! We've successfully navigated the world of coordinate geometry, used the slope-intercept form, and found the value of 'a'. From slopes and y-intercepts to the final solution, we've gone on a journey that demonstrates the interconnectedness of all the points on a line. It's a victory for anyone looking to sharpen their skills. Keep practicing, and you'll become a pro in no time.

The Importance of Practice

Mathematics, like any skill, requires practice. The more you solve problems like these, the more comfortable and confident you'll become. Each problem you solve is a step forward, solidifying your understanding and building your problem-solving muscle. Don't be afraid to make mistakes; they are an essential part of the learning process. Embrace the challenge, and enjoy the journey of discovery.

So next time you encounter a similar problem, remember this guide, and remember that with the right tools and a little persistence, you can conquer any mathematical challenge. Keep exploring, keep learning, and keep the curiosity alive! The world of mathematics awaits, full of exciting discoveries and intellectual rewards.