Finding Equations With A Specific Slope And Point

by SLV Team 50 views

Hey guys! Let's dive into a common math problem: finding the equation of a line when you know its slope and a point it passes through. This is super useful, whether you're working on a math assignment or trying to understand real-world relationships. We'll break down the concepts, and then look at the given multiple-choice options to figure out which one fits the bill. Are you ready?

Understanding the Basics: Point-Slope Form

Alright, first things first. To tackle this, we need to understand the point-slope form of a linear equation. This is the MVP (Most Valuable Player) of this type of problem. The point-slope form is written as: y - y₁ = m(x - x₁).

  • Where:
    • m represents the slope of the line (how steep it is).
    • (x₁, y₁) represents a specific point that the line passes through.

So, if we're given the slope and a point, we can just plug those values into this formula, and we're good to go! Easy peasy, right?

Let's relate the given information to our problem. We're told the slope is 5.9. That's our 'm'. We are also told that the line passes through the point (-28, 47). This means x₁ = -28 and y₁ = 47. With these values, we can then rewrite the point-slope form as: y - 47 = 5.9(x - (-28)). Simplifying it, we get y - 47 = 5.9(x + 28). Let's use this understanding to analyze the provided options. This knowledge is fundamental and acts as the foundation of our solution. In essence, the point-slope form provides a direct method to derive the equation of a line, given its slope and a specific point. The beauty of this form is its straightforward application; you only need to substitute the given values to arrive at the solution. The ability to transform given information into the point-slope form simplifies the complex process of linear equation derivation, making it an invaluable tool in mathematical problem-solving. This formula serves as a core principle for understanding the relationship between the slope, points, and the linear equation representing a line. Getting familiar with it makes solving linear equations a whole lot easier.

Analyzing the Options: Which Equation Works?

Now, let's look at the multiple-choice options. Remember, we're looking for an equation that has a slope of 5.9 and passes through the point (-28, 47). We've already established how this should look, so let's compare our findings to the choices. Remember that our target equation looks like this: y - 47 = 5.9(x - (-28)), which simplifies to y - 47 = 5.9(x + 28). So we're looking for this exact equation, or an equivalent one. It's really that simple.

Let's go through the list:

  1. y - 47 = 5.9(x - 28): This is incorrect. The point doesn't align with the expected form, as the x-coordinate should be added, not subtracted.
  2. y + 47 = 5.9(x + 28): Incorrect. The y-coordinate has the wrong sign.
  3. y + 47 = 5.9(x - 28): Incorrect. The y-coordinate and the x-coordinate have the wrong sign.
  4. y - 47 = 5.9(x + 28): This is the correct answer! This is exactly what we derived from the point-slope form.
  5. y + 28 = 5.9(x - 47): Incorrect. The slope is correct, but the point does not fit.
  6. y + 28 = 5.9(x + 47): Incorrect. The slope is correct, but the point does not fit.
  7. y - 28 = 5.9(x - 47): Incorrect. The y-coordinate has the wrong sign, as well as x-coordinate.
  8. y - 28 = 5.9(x + 47): Incorrect. The y-coordinate has the wrong sign, as well as x-coordinate.

So, as we can see, the equation that perfectly matches our criteria is the one where the y-coordinate is subtracted (47) and the x-coordinate is added (28), and the whole expression is multiplied by the slope (5.9). The process of elimination is a good strategy to approach these kinds of problems, as it helps narrow down to the right solution. In essence, the proper application of the point-slope form, and a thorough comparison against the options, is key to this type of problem. It's also important to remember the signs to be certain that the x and y coordinates are correctly associated with the slope value.

Let's Simplify: Understanding the Signs

I want to focus a bit more on how to quickly spot the right answer in these multiple-choice problems. The most common mistake is mixing up the signs of the coordinates. The point-slope form uses subtraction. So, when you substitute the coordinates of your point into the equation, you need to be very careful with the signs. Here's a quick trick:

  • If the x-coordinate is negative (like -28), the equation will have a plus sign (+28) inside the parentheses: (x + 28)
  • If the y-coordinate is positive (like 47), the equation will have a minus sign (-47) outside the parentheses: (y - 47)

This simple understanding helps you avoid the common pitfalls and quickly identify the correct equation. Always cross-check the signs and you'll be golden. The proper way of thinking about this is: the point slope form wants the opposite of the coordinate signs. The point-slope form is designed to provide you with a systematic method to create linear equations; knowing how the signs operate helps to keep the solution consistent. The proper application of these sign rules is fundamental, and it will prevent you from accidentally selecting the wrong solution.

Wrapping Up: Putting It All Together

So there you have it, guys! We've found the equation that has a slope of 5.9 and passes through the point (-28, 47). We've also learned the importance of the point-slope form and how to apply it, and we've discussed how the signs play an important role. Remember, math is all about understanding the concepts and applying them. Keep practicing, and you'll become a pro at these types of problems! I hope you found this guide helpful. If you have any more questions, just let me know. Good luck, and keep learning!