Is (-1,-4) A Solution To 5x + 9y = 5?

Hey guys, let's dive into a super common math problem that pops up a lot in algebra: figuring out if a specific point, like (-1, -4), actually works as a solution for an equation, in this case, 5x + 9y = 5. It might sound a little intimidating at first, but trust me, it's all about plugging in the numbers and seeing if things balance out. We're basically playing detective with numbers here, checking if our potential solution fits the crime scene – which is our equation! So, grab your thinking caps, and let's break down exactly how we do this. This skill is fundamental, and once you get the hang of it, you'll be spotting solutions like a pro. We'll go through it step-by-step, making sure you understand why we do each part. This isn't just about getting the right answer; it's about understanding the logic behind it. Ready to solve this mathematical mystery?

Understanding What a Solution Means

So, what does it actually mean for a point like (-1, -4) to be a solution to an equation like 5x + 9y = 5? Think of an equation as a statement of balance. The left side of the equals sign (in this case, 5x + 9y) must be exactly equal to the right side (which is 5). When we talk about a point being a solution, we're saying that if you substitute the x-value and the y-value of that point into the equation, the balance holds true. For our point (-1, -4), the first number, -1, is the x-coordinate, and the second number, -4, is the y-coordinate. If (-1, -4) is indeed a solution, then when we plug x = -1 and y = -4 into the equation 5x + 9y = 5, the entire statement should become true. It's like checking if a key fits a lock; if it's the right key (the solution), the lock (the equation) will open (be true). If the key doesn't fit, the lock remains shut (the equation is false). This concept is super important because it's the foundation for solving systems of equations, graphing lines, and so much more in the world of mathematics. We're not just randomly trying points; we're verifying if they satisfy the conditions set by the equation. This verification process is a core skill in algebra and beyond.

The Substitution Method: Plugging In the Values

Alright, let's get down to business with the substitution method. This is our main tool for checking if (-1, -4) is a solution to 5x + 9y = 5. It's pretty straightforward, guys. We take our equation and our point and carefully replace the variables with their corresponding values. Remember, in the point (-1, -4), the first number is x and the second number is y. So, we're going to substitute x = -1 and y = -4 into the equation 5x + 9y = 5. Here’s how it looks:

• Start with the original equation: $5x + 9y = 5$
• Identify the values to substitute: $x = -1$ and $y = -4$
• Substitute these values into the equation: Replace every 'x' with '(-1)' and every 'y' with '(-4)'. It's crucial to use parentheses, especially when dealing with negative numbers, to avoid making sign errors. This is where many people can trip up, so pay close attention!

So, our equation becomes:

$5(-1) + 9(-4) = 5$

See? We've replaced 'x' with '(-1)' and 'y' with '(-4)'. Now, the next step is to simplify the left side of the equation and see if it actually equals the right side. This is where the magic happens, and we find out if our point is a true solution or not. It's all about careful calculation from here on out. We'll tackle that in the next section.

Calculation and Verification: Does it Balance?

Now for the exciting part: the calculation and verification! We've already done the heavy lifting by substituting x = -1 and y = -4 into the equation 5x + 9y = 5, giving us: $5(-1) + 9(-4) = 5$. The next step is to perform the arithmetic operations on the left side of the equation to see if it simplifies to 5. Let's break it down:

• First multiplication: $5 imes (-1) = -5$. Remember, a positive number multiplied by a negative number results in a negative number.
• Second multiplication: $9 imes (-4) = -36$. Again, a positive number multiplied by a negative number gives a negative result.

Now, substitute these results back into our equation:

$-5 + (-36) = 5$

Adding a negative number is the same as subtracting its positive counterpart. So, we can rewrite this as:

$-5 - 36 = 5$

Now, let's perform the subtraction (or addition of negative numbers):

$-5 - 36 = -41$

So, the left side of our equation simplifies to -41. Our original equation was 5x + 9y = 5. After substituting our point (-1, -4), we found that the left side equals -41, while the right side is 5.

Is -41 equal to 5? Absolutely not!

This means that when we plug in $x = -1$ and $y = -4$, the equation 5x + 9y = 5 does not hold true. Therefore, the point (-1, -4) is not a solution to the equation $5x + 9y = 5$.

Conclusion: The Verdict on (-1,-4)

So, after all our detective work, plugging in the numbers, and doing the calculations, we've reached our conclusion. We tested the point (-1, -4) in the equation 5x + 9y = 5. We substituted $x = -1$ and $y = -4$ into the equation, which gave us $5(-1) + 9(-4)$. Performing the multiplication, we got $-5 + (-36)$. Adding these together, we found the left side of the equation simplifies to $-41$. Since $-41$ is not equal to the right side of the equation, which is $5$, the point (-1, -4) does not satisfy the equation. Therefore, guys, (-1, -4) is NOT a solution of $5x + 9y = 5$. It's that simple! It's all about the verification process. If the numbers match up and the equation is true, then it's a solution. If they don't match, then it's not. Keep practicing this, and you'll become a math whiz in no time!