Solve Equations Find Ordered Pair Solutions For Y-4=7(x-6)

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Hey guys! Let's dive into a common math problem where we need to figure out which ordered pair makes an equation true. We'll break down the equation y - 4 = 7(x - 6), and test some ordered pairs to see if they're solutions. It's like a detective game, but with numbers!

Understanding the Equation y - 4 = 7(x - 6)

Okay, so first things first, let's really understand the equation y *- 4 = 7(x - 6). In this equation, we're dealing with two variables, x and y. Think of it like a secret relationship between these two numbers. The equation tells us exactly how they're connected. The heart of understanding this linear equation lies in recognizing that it represents a straight line when graphed on a coordinate plane. Every point (x, y) on this line is a solution to the equation, meaning when you plug those x and y values into the equation, it balances out perfectly. This form of the equation is actually a variation of the point-slope form, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. Spotting this connection can give you a quick insight into the equation's properties. But how do we find these magical points? That's where ordered pairs come in. An ordered pair, like (5, 4) or (6, 5), is simply a pair of numbers where the order matters. The first number is the x-coordinate, and the second number is the y-coordinate. Our mission is to see if any of the given ordered pairs fit into our equation's secret relationship. We do this by substituting the x and y values from the ordered pair into the equation and checking if both sides are equal. If they are, bingo! We've found a solution. If not, well, the pair just wasn't meant to be a part of our equation's story. So, in a nutshell, we're not just solving for a single number here; we're finding pairs of numbers that work together in a specific way defined by the equation. This concept is super important in algebra and beyond, so let's get good at it!

Testing Ordered Pair A: (5, 4)

Let's test the first ordered pair, (5, 4), to see if it’s a solution for the equation y *- 4 = 7(x - 6). Remember, an ordered pair is written as (x, y), so in this case, x = 5 and y = 4. The crucial step here is substitution. We're going to take these values and plug them directly into our equation, replacing the variables. So, everywhere we see a y, we'll put a 4, and everywhere we see an x, we'll put a 5. Our equation now looks like this: 4 - 4 = 7(5 - 6). Now, it's all about simplifying and seeing if both sides of the equation end up being equal. On the left side, we have 4 - 4, which is simply 0. Easy peasy! On the right side, we have 7(5 - 6). First, we need to tackle what's inside the parentheses: 5 - 6. That gives us -1. So now we have 7(-1). Multiplying 7 by -1, we get -7. So, our equation now looks like this: 0 = -7. Hold on a second! Does 0 equal -7? Nope, it definitely doesn't. This means that the ordered pair (5, 4) does not satisfy the equation. It's not a solution. Think of it like trying to fit the wrong puzzle piece – it just won't work. So, we've learned something important here: not every ordered pair is going to be a solution. We need to find the pairs that make the equation a true statement. This is why testing each pair is so important. We can't just guess; we have to do the math and see if it checks out. Now that we've seen (5, 4) doesn't work, let's move on to the next contender and see if it fares any better!

Testing Ordered Pair B: (6, 5)

Alright, let's put the ordered pair (6, 5) to the test and see if it’s a solution for our equation y *- 4 = 7(x - 6). Just like before, we know that in the ordered pair (6, 5), x is 6 and y is 5. We're going to use the same method of substitution as we did with the first pair. This means we'll replace the y in the equation with 5 and the x with 6. Our equation now transforms into: 5 - 4 = 7(6 - 6). Time to simplify and see if both sides balance out. On the left side, we've got 5 - 4, which is a straightforward subtraction. 5 minus 4 equals 1. So the left side of our equation is simply 1. Now, let's tackle the right side: 7(6 - 6). The first thing we need to do is simplify what's inside the parentheses: 6 - 6. And that gives us 0. So now the right side of our equation looks like 7(0). Anything multiplied by 0 is 0, so 7 times 0 is 0. Our equation now reads: 1 = 0. Hmmm, does 1 equal 0? Nope, not in the world of mathematics! This tells us that the ordered pair (6, 5) is not a solution to the equation y - 4 = 7(x - 6). It didn't make the equation true. Just like the first ordered pair, it doesn't fit the relationship defined by our equation. So, what does this mean? It means we're building a clearer picture of what kind of ordered pairs will work. We've seen two that don't, which helps us narrow down the possibilities. The key takeaway here is that each ordered pair needs to be rigorously tested by plugging its values into the equation and verifying if it holds true. With (6, 5) ruled out, we're ready to consider the other options, keeping in mind that we're looking for the pair, or pairs, that make the equation sing!

Determining the Correct Answer: Neither (5, 4) nor (6, 5)

So, we've put both ordered pairs, (5, 4) and (6, 5), through the wringer and neither of them turned out to be a solution for the equation y *- 4 = 7(x - 6). Remember, when we tested (5, 4), we ended up with the equation 0 = -7, which is definitely not true. And when we tested (6, 5), we got 1 = 0, which is also a false statement. This means that neither of these pairs satisfies the equation; they don't fit the relationship between x and y that the equation describes. What does this tell us about the answer choices? Let's break them down:

  • A. Only (5, 4): We know this is incorrect because (5, 4) didn't work.
  • B. Only (6, 5): This is also incorrect since (6, 5) wasn't a solution either.
  • C. Both (5, 4) and (6, 5): This is wrong because we've confirmed that neither pair works.
  • D. Neither: This is looking like our winner! Since both pairs failed the test, the correct answer is that neither (5, 4) nor (6, 5) is a solution to the equation. This process of elimination is a powerful tool in math. By systematically testing each option, we can confidently arrive at the correct answer, even if it's