Finding Ordered Pair Solutions For Equation Y - 4 = 7(x - 6)

Hey guys! Let's dive into solving this equation and figuring out which ordered pair is the real deal. This is a super common type of problem in algebra, and once you get the hang of it, you'll be solving these like a pro. We're given the equation y - 4 = 7(x - 6), and we need to determine which of the provided ordered pairs, if any, satisfy this equation. Ordered pairs are just coordinate points (x, y), so we're going to plug in the x and y values from each option into the equation and see if it balances out. If it does, we've found our solution! If not, we move on to the next one. Remember, the goal here is to find the pair(s) that make the equation true. So, let's get started and break down each step to make sure we understand exactly how to solve this. Understanding how to manipulate equations and substitute values is a fundamental skill in mathematics, and it opens the door to more complex problem-solving later on. Think of this as building the foundation for your math superpowers! We'll take a look at each option, carefully substituting the values and simplifying to see if the equation holds true. It's like being a detective, where we're searching for the clues that fit the puzzle perfectly. And don't worry if it seems a bit tricky at first; with a little practice, it will become second nature. Let's jump in and see what we can uncover!

Checking Option A: Only (5, 4)

Okay, let's start by checking the first option, ordered pair (5, 4). This means x = 5 and y = 4. We're going to substitute these values into our equation y - 4 = 7(x - 6) and see if both sides of the equation are equal. It's like trying to fit a key into a lock; if it fits, we've got a solution! So, let's plug in the numbers:

4 - 4 = 7(5 - 6)

First, we simplify both sides independently. On the left side, 4 - 4 equals 0. On the right side, we first deal with the parentheses: 5 - 6 equals -1. So now our equation looks like this:

0 = 7(-1)

Next, we multiply 7 by -1, which gives us -7. So now we have:

0 = -7

This statement is clearly false. 0 does not equal -7. This means that the ordered pair (5, 4) does not satisfy the equation. It's like trying to fit a square peg in a round hole; it just doesn't work! So, we can rule out option A as the correct answer. But don't worry, we've still got other options to check. This is a process of elimination, and we're making progress by ruling out incorrect answers. Keep in mind that this is a crucial step in problem-solving – not every guess will be right, and that's okay! We learn from each attempt and move closer to the correct solution. Now, let's move on and check the next ordered pair. Who knows? It might be the one we're looking for!

Checking Option B: Only (6, 5)

Alright, let's move on to the next contender: ordered pair (6, 5). This time, we have x = 6 and y = 5. Just like before, we're going to substitute these values into our trusty equation y - 4 = 7(x - 6) and see if the equation holds true. Think of it as putting our detective hats back on and looking for more clues! Let's plug in the values:

5 - 4 = 7(6 - 6)

Again, we'll simplify each side of the equation. On the left side, 5 - 4 equals 1. On the right side, we first handle the parentheses: 6 - 6 equals 0. So, our equation now looks like this:

1 = 7(0)

Now, we multiply 7 by 0, which gives us 0. Our equation becomes:

1 = 0

This statement is also false. 1 does not equal 0. So, the ordered pair (6, 5) does not satisfy the equation either. It's like striking another match and it not lighting; we keep trying! This means we can eliminate option B as well. Don't get discouraged; sometimes you have to go through a few tries before you find the right answer. The key is to be methodical and carefully check each option. We're learning more with each step, and that's what matters. We've ruled out two possibilities so far, which means we're getting closer to the solution. Now, let's move on to the next option and see if it's the winner!

Checking Option C: Both (5, 4) and (6, 5)

Now, let's consider option C, which suggests that both ordered pairs (5, 4) and (6, 5) are solutions. But hold on a second! Remember what we found when we checked options A and B? We discovered that neither (5, 4) nor (6, 5) individually satisfied the equation y - 4 = 7(x - 6). Think back to our detective work; we already know these clues don't fit! So, if neither pair works on its own, it's impossible for both of them to be solutions. This is a crucial logical step in problem-solving. We've already done the hard work of checking each pair individually, so we can confidently rule out this option. It's like knowing that two puzzle pieces don't fit separately, so they definitely won't fit together! Option C is essentially trying to trick us by combining two incorrect answers. This is a common tactic in multiple-choice questions, so it's important to stay sharp and use the information we've already gathered. We're using our previous results to make an informed decision, and that's a sign of strong problem-solving skills. So, with a clear head, we can confidently say that option C is not the correct answer. We're narrowing down the possibilities, and that's a great feeling. Let's move on to the final option and see if it's the solution we've been searching for!

Determining Option D: Neither

We've reached the final option, D: Neither. We've already thoroughly checked options A, B, and C. We found that (5, 4) is not a solution, (6, 5) is not a solution, and therefore, the option claiming both are solutions is also incorrect. It's like we've eliminated all the suspects except for one! So, by the process of elimination, if none of the other options are correct, then option D, Neither, must be the correct answer. This is a powerful technique in problem-solving, especially in multiple-choice situations. When you've carefully ruled out all other possibilities, you can be confident that the remaining option is the correct one. Think of it as closing the case; we've gathered all the evidence and reached our conclusion! In this case, we've systematically checked each ordered pair and shown that they do not satisfy the equation y - 4 = 7(x - 6). Therefore, option D, Neither, is indeed the solution. This is a satisfying moment – we've solved the puzzle! It's important to remember that sometimes the answer is "none of the above," and that's perfectly valid. We shouldn't be afraid to choose that option if it's the logical conclusion. And guys, that's how it's done! We've successfully navigated this problem by carefully substituting values, simplifying equations, and using the process of elimination. You've got this!

Final Answer

So, the correct answer is D. Neither. Great job, everyone! Keep practicing, and you'll become even more confident in your algebra skills.