Finding The Equation Of A Line From Given Points

Hey guys! Let's dive into a cool math problem today. We're given some points that lie on a straight line: (-2, -6), (1, 3), (3, 9), and (4, 12). Our mission is to figure out which equation represents this line from the options provided: a. x + y - 2 = 0, b. 3x - 2y - 6 = 0, and c. 2x + 3y + 6 = 0. It might sound tricky, but trust me, we'll break it down and make it super easy to understand. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we really understand what's going on. We know that all these points (-2, -6), (1, 3), (3, 9), and (4, 12) sit perfectly on a single, straight line. That's the key piece of information here. Each of these points has an x-coordinate and a y-coordinate, and these coordinates follow a specific pattern because they're on the same line. This pattern is what we need to capture in an equation. Think of a line like a secret code that connects all these points. Our job is to crack that code and find the equation that describes it. The equations given (x + y - 2 = 0, 3x - 2y - 6 = 0, and 2x + 3y + 6 = 0) are all linear equations, meaning they represent straight lines. We need to figure out which one fits our set of points. To do this, we'll use a simple but powerful method: plugging in the coordinates and seeing which equation holds true for all the points. This is where the fun begins!

Method 1: Testing the Points in Each Equation

Okay, so the first method we're going to use is pretty straightforward: we'll take each point and plug its x and y values into each equation. If the equation holds true (meaning both sides are equal) for all the points, then we've found our winner. This method is all about testing and checking, so let's get started. First, let's look at equation a: x + y - 2 = 0. We'll plug in the first point, (-2, -6), which means x = -2 and y = -6. So, -2 + (-6) - 2 = -10. This does not equal 0, so equation a doesn't work for this point. Since it doesn't work for even one point, we can rule it out immediately. No need to waste time testing the other points! Next up is equation b: 3x - 2y - 6 = 0. Let's plug in (-2, -6) again. 3*(-2) - 2*(-6) - 6 = -6 + 12 - 6 = 0. Okay, it works for the first point! But remember, it needs to work for all the points. Let's try the second point, (1, 3). 3*(1) - 2*(3) - 6 = 3 - 6 - 6 = -9. This does not equal 0, so equation b is also out. Finally, let's try equation c: 2x + 3y + 6 = 0. Plugging in (-2, -6), we get 2*(-2) + 3*(-6) + 6 = -4 - 18 + 6 = -16. This also doesn't equal 0, so equation c doesn't seem to be the right one either, at least not directly. Hmmm, this is interesting! It looks like none of the given equations fit the points perfectly. Maybe we need to rethink our approach or double-check the problem. Don't worry, this is common in problem-solving. Sometimes the first try doesn't give us the answer, but it gives us valuable clues.

Method 2: Finding the Slope and Y-intercept

Okay guys, since directly plugging in the points didn't give us an immediate answer, let's switch gears and try a different approach. This time, we'll focus on finding the slope and y-intercept of the line. Remember, every straight line can be described by the equation y = mx + b, where 'm' is the slope and 'b' is the y-intercept. If we can find these two values, we can construct the equation of the line. First, let's calculate the slope. The slope (m) is the measure of how steep the line is, and we can find it using any two points on the line. The formula for slope is m = (y2 - y1) / (x2 - x1). Let's use the points (-2, -6) and (1, 3) for this calculation. Plugging in the values, we get m = (3 - (-6)) / (1 - (-2)) = (3 + 6) / (1 + 2) = 9 / 3 = 3. So, the slope of our line is 3. That's one piece of the puzzle solved! Now, we need to find the y-intercept (b). This is the point where the line crosses the y-axis (where x = 0). To find 'b', we can use the slope we just calculated and one of the points on the line. Let's use the point (1, 3) and plug it into the equation y = mx + b. We have 3 = 3*(1) + b. Simplifying, we get 3 = 3 + b. Subtracting 3 from both sides, we find that b = 0. So, the y-intercept is 0. Now we have both the slope (m = 3) and the y-intercept (b = 0). We can plug these values into the equation y = mx + b to get the equation of the line: y = 3x + 0, which simplifies to y = 3x. But wait! This equation isn't in the same form as the options given. We need to rearrange it to match one of the options. Let's subtract 3x from both sides to get -3x + y = 0. Multiplying the entire equation by -1, we get 3x - y = 0. Now, let's compare this to the given options. It still doesn't match exactly, but we're getting closer! Notice that if we rearrange equation b (3x - 2y - 6 = 0), it looks somewhat similar. This suggests we might need to manipulate our equation further or re-examine the options.

Method 3: Comparing Slopes and Forms

Alright, let's try a third method to make sure we're on the right track. This time, we'll focus on comparing the slopes and the general form of the equations. We already found that the slope of the line passing through our points is 3. This is a crucial piece of information because it narrows down our options. Now, let's rewrite the given equations in the slope-intercept form (y = mx + b) to easily see their slopes. Equation a: x + y - 2 = 0 can be rewritten as y = -x + 2. The slope here is -1, which doesn't match our calculated slope of 3, so we can rule out option a. Equation b: 3x - 2y - 6 = 0 can be rewritten as 2y = 3x - 6, and then y = (3/2)x - 3. The slope here is 3/2, which also doesn't match our slope of 3, so option b is out as well. Equation c: 2x + 3y + 6 = 0 can be rewritten as 3y = -2x - 6, and then y = (-2/3)x - 2. The slope here is -2/3, which again doesn't match our slope of 3. It seems like none of the provided equations perfectly fit the points when we look at the slope-intercept form. However, let's think about what we've found. We know the line has a slope of 3 and passes through the point (0,0). The equation y=3x represents a line that passes the criteria. Let's see how we can relate our base equation to the options. Sometimes, a multiple of the equation can also represent the same line if it satisfies the initial conditions. Let's revisit our initial strategy of plugging in points. We might have missed something in the arithmetic, or there might be a clever way to manipulate the equations.

Conclusion

Okay, guys, we've explored a few different methods to tackle this problem, and it seems like none of the given options perfectly match the equation of the line we found (y = 3x). This can happen sometimes, and it's a good reminder that not every problem has a straightforward answer right away. We double-checked our calculations for the slope and y-intercept, and we're confident that y = 3x is indeed the equation representing the line passing through the given points. However, none of the options (a, b, or c) directly translate to this equation. What does this mean? Well, there could be a few possibilities. Maybe there was a slight error in the original problem statement, or perhaps the options provided were designed to be a bit of a trick. It's also possible that there's a way to manipulate the equations further that we haven't considered yet, but as it stands, our analysis suggests that none of the given options are correct. The key takeaway here is that problem-solving is a journey. Sometimes you reach a dead end, but that doesn't mean you've failed. It just means it's time to re-evaluate, try a different approach, or even double-check the initial information. We've learned a lot by working through this problem, even if we didn't arrive at one of the multiple-choice answers. We reinforced our understanding of slopes, y-intercepts, and how to manipulate linear equations. And that, my friends, is a win in itself! Keep practicing, keep exploring, and never be afraid to try different angles when tackling a challenge.