Solving For B: -x + By = 10 With Point (-2, -2)

Hey guys! Today, we're diving into a fun little math problem where we need to solve for B in the equation -x + By = 10, and we're given the point (-2, -2). This means we know a specific x and y value that makes this equation true. Think of it like a puzzle where we have most of the pieces, and we just need to find the missing one, which in this case is B. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand what the problem is asking. We have a linear equation, -x + By = 10. This equation represents a line, and B is a coefficient that affects the slope of this line. We're also given a point, (-2, -2). This point lies on the line, meaning that if we plug these values for x and y into the equation, it should hold true. Our mission is to find the value of B that makes this happen. Essentially, we're finding the specific line (out of many possible lines) that passes through the point (-2, -2) and satisfies the equation's form. This kind of problem is a classic example of using given information to find an unknown variable, a fundamental skill in algebra and beyond. We use this concept in various real-world applications, from engineering to economics, where we often need to determine specific parameters based on observed data or conditions. So, let's break down the steps and solve for B like pros!

Step-by-Step Solution

Okay, let's break down how to solve for B step-by-step. This is where the fun begins! Remember, our equation is -x + By = 10, and our point is (-2, -2). This means x = -2 and y = -2. We’re going to substitute these values into the equation. Think of it like replacing the letters with their numerical values, like swapping players on a sports team. This substitution is the key to unlocking the value of B. Once we've substituted, we'll have a simpler equation with just B as the unknown, making it much easier to isolate and solve. So, let’s get into the nitty-gritty and see how it works!

1. Substitute the values of x and y

The first thing we need to do is substitute the given values of x and y into our equation. This is like plugging in the coordinates of a specific location on a map into our route. We know that x = -2 and y = -2, and our equation is -x + By = 10. So, let's replace x with -2 and y with -2 in the equation. This gives us:

-(-2) + B(-2) = 10

See how we've simply swapped the letters x and y with their corresponding numbers? Now we have an equation that only involves B, which is exactly what we want! This step is crucial because it transforms the equation into a solvable form. We've taken the first big step in our quest to solve for B. Next, we'll simplify this equation to make it even easier to handle. Onward and upward!

2. Simplify the equation

Alright, now that we've substituted the values, it's time to simplify the equation. This is like tidying up our workspace before we get to the main task. Our equation currently looks like this: -(-2) + B(-2) = 10. First, let's deal with the double negative. Remember, a negative times a negative is a positive, so -(-2) becomes 2. Now our equation looks like this:

2 + B(-2) = 10

Next, let's simplify B(-2). We can rewrite this as -2B. So, our equation now looks even cleaner:

2 - 2B = 10

We've made some serious progress! By simplifying, we've made the equation much easier to work with. It's like taking a tangled mess of yarn and turning it into a neat ball, ready to be used. Now, we're one step closer to isolating B and finding its value. The next step is to isolate the term with B on one side of the equation. Let's keep the momentum going!

3. Isolate the term with B

Great job so far! We've simplified our equation to 2 - 2B = 10. Now, we need to isolate the term with B, which is -2B. Think of isolating a variable like separating a specific ingredient in a recipe so you can measure it properly. To isolate -2B, we need to get rid of the 2 on the left side of the equation. We can do this by subtracting 2 from both sides of the equation. Remember, what we do to one side, we must do to the other to keep the equation balanced. It’s like a seesaw – if you add weight to one side, you need to add the same weight to the other to keep it level.

So, let's subtract 2 from both sides:

2 - 2B - 2 = 10 - 2

This simplifies to:

-2B = 8

Excellent! We've successfully isolated the term with B. It’s like we’ve cornered our target and now we can zero in on finding its exact value. We’re almost there! The last step is to solve for B itself. Let’s move on to the final piece of the puzzle.

4. Solve for B

We're in the home stretch now! We've isolated the term with B, and our equation is -2B = 8. To finally solve for B, we need to get B all by itself on one side of the equation. Currently, B is being multiplied by -2. To undo this multiplication, we need to divide both sides of the equation by -2. Remember, division is the inverse operation of multiplication, just like subtraction is the inverse of addition. So, let’s divide both sides by -2:

-2B / -2 = 8 / -2

This simplifies to:

B = -4

Boom! We did it! We've successfully solved for B. It’s like reaching the final destination on a treasure map. B is equal to -4. This means that the value of B that makes the equation -x + By = 10 true for the point (-2, -2) is -4. Give yourselves a pat on the back – you’ve conquered this algebraic challenge!

Checking Our Answer

Now that we've found B = -4, it's always a good idea to check our answer. This is like double-checking your work on a test to make sure you didn't make any silly mistakes. To check, we'll substitute B = -4, x = -2, and y = -2 back into our original equation, -x + By = 10, and see if it holds true. Let's plug in the values:

-(-2) + (-4)(-2) = 10

Now, let's simplify:

2 + 8 = 10

10 = 10

Woohoo! The equation holds true! This confirms that our answer, B = -4, is correct. Checking our answer is a crucial step because it helps us catch any errors and ensures that we have the right solution. It’s like having a safety net – it gives us confidence that we’ve done the problem correctly. So, never skip this step, guys! It can save you a lot of headaches in the long run.

Conclusion

Alright, guys, we've done it! We successfully solved for B in the equation -x + By = 10, given the point (-2, -2). We found that B = -4. We went through each step carefully, from substituting the values of x and y, simplifying the equation, isolating the term with B, solving for B, and finally, checking our answer. This problem is a great example of how we can use algebraic principles to find unknown variables. Remember, the key is to break down the problem into smaller, manageable steps, and to always double-check your work. You guys are awesome, and I’m super proud of how we tackled this problem together! Keep practicing, and you'll become math masters in no time! Now go forth and conquer more math challenges!