Does (1,1) Satisfy Y=4x-5?

Hey guys, let's dive into a quick math problem that's super common in algebra. We're looking at whether the ordered pair $(1,1)$ is a valid solution to the equation $y = 4x - 5$. It sounds simple, and honestly, it is once you know the trick! This concept is fundamental when you're first learning about graphing lines and understanding what it means for a point to 'be on' a line. So, grab your favorite beverage, get comfy, and let's break this down.

Understanding Ordered Pairs and Equations

First off, let's chat about what an ordered pair like $(1,1)$ actually means. In mathematics, an ordered pair is a set of two numbers written in a specific order, usually in parentheses and separated by a comma. The first number, which is $1$ in our case, represents the $x$-coordinate, and the second number, also $1$ here, represents the $y$-coordinate. Think of it like coordinates on a map – the order matters! You wouldn't want to tell someone to go "1 mile east, then 1 mile north" and have them go "1 mile north, then 1 mile east"; it gets you to the same spot, but in math, we're often very precise about the order. This ordered pair $(1,1)$ is essentially a specific point on a graph.

Now, let's talk about the equation $y = 4x - 5$. This equation describes a relationship between $x$ and $y$. It's a linear equation, meaning if you were to plot all the points $(x,y)$ that satisfy this equation, they would form a straight line on a graph. The equation tells us how $x$ and $y$ are related. For every value of $x$ you plug in, there's a corresponding value of $y$ that makes the equation true. Our job is to see if the specific point $(1,1)$ fits this relationship. Does this particular pair of $x$ and $y$ values make the equation a true statement? That's the million-dollar question (or, you know, the single-point question).

So, to determine if an ordered pair is a solution to an equation, we simply substitute the $x$ and $y$ values from the ordered pair into the equation. If the equation holds true – meaning the left side equals the right side after the substitution – then the ordered pair is indeed a solution. If it doesn't balance out, then that point isn't on the line represented by the equation. It's like a key fitting into a lock; if it works, great, if not, you need a different key (or in this case, a different point!). This process is super important for checking your work when solving systems of equations or verifying points on a curve. It’s a fundamental skill that underpins a lot of higher-level math.

The Substitution Test

Alright guys, let's get hands-on with the substitution. We have our ordered pair $(1,1)$, which means $x=1$ and $y=1$. Our equation is $y = 4x - 5$. The game plan here is simple: replace every 'y' in the equation with $1$ and every 'x' with $1$. It's like a direct swap. So, the equation becomes:

$1 = 4(1) - 5$

See what we did there? We took the $y$ from the ordered pair and put it on the left side, and we took the $x$ from the ordered pair and put it inside the parentheses on the right side. This is the crucial step. Now, we just need to simplify the right side of the equation and see if it equals the left side.

Let's do the math:

$4(1)$ is simply $4$. So, the equation is now:

$1 = 4 - 5$

Next, we perform the subtraction on the right side: $4 - 5$. This gives us $-1$. So, the equation simplifies to:

$1 = -1$

Now, we look at this final statement. Is $1$ equal to $-1$? Absolutely not! These are two different numbers. The left side ($1$) does not equal the right side ($-1$). This means that when we plug in the values $x=1$ and $y=1$ into the equation $y = 4x - 5$, the equation does not hold true. It results in a false statement.

What Does This Mean?

Because the equation $1 = -1$ is false, the ordered pair $(1,1)$ is not a solution to the equation $y = 4x - 5$. This point $(1,1)$ does not lie on the line represented by this equation. If you were to graph the line $y = 4x - 5$, the point $(1,1)$ would be somewhere else on the graph, not on the line itself. It's like trying to place a puzzle piece in the wrong spot – it just doesn't fit the picture defined by the equation. This might seem a bit disappointing if you were hoping it was a solution, but understanding why it isn't is just as important as knowing if it is. It reinforces the concept of what it means for a point to satisfy an equation.

The Importance of Checking Your Solutions

Guys, this process of checking solutions is a cornerstone of mathematics, especially when you're tackling more complex problems. Imagine you've spent ages solving a system of equations, and you finally get a potential solution, say an ordered pair like $(2,3)$. How do you know if you're right? You use the substitution test we just did! You plug $(2,3)$ into both original equations. If it satisfies both, then you've likely nailed it. If it only satisfies one, or neither, you know you need to go back and review your steps. This verification step is your safety net, preventing you from going down the wrong path with incorrect answers.

In the context of the problem $y = 4x - 5$ and the point $(1,1)$, we found that $1 eq -1$. This definitively tells us that $(1,1)$ is not a solution. The correct answer to whether $(1,1)$ is a solution is therefore False. It's crucial to trust the math here. The substitution method is an objective way to verify results. It doesn't rely on guesswork or intuition; it relies on the fundamental rules of arithmetic and algebra.

Beyond Simple Substitution

Thinking a bit deeper, what would be a solution to $y = 4x - 5$? Let's pick an $x$ value, say $x=2$. Then $y = 4(2) - 5 = 8 - 5 = 3$. So, the ordered pair $(2,3)$ is a solution because $3 = 4(2) - 5$ simplifies to $3 = 8 - 5$, which is $3=3$. This is true! So, the point $(2,3)$ lies on the line. This comparison highlights the difference between a point that satisfies the equation and one that doesn't. When you're working with graphs, every point on the line is a solution, and every point off the line is not. Understanding this relationship is key to visualizing and interpreting mathematical relationships.

Furthermore, this principle extends to functions and more intricate mathematical models. Whether you're dealing with quadratic equations, trigonometric functions, or calculus problems, the concept of verifying solutions by substitution remains a powerful tool. It's the bedrock upon which mathematical accuracy is built. So, next time you're faced with a similar question, remember the simple substitution test. It’s your trusty sidekick in the world of math problem-solving. It might seem basic, but mastering these fundamentals is what allows you to confidently tackle the more challenging aspects of mathematics later on. Keep practicing, and you'll be a substitution pro in no time!

Conclusion: The Verdict

So, to wrap things up, we took the ordered pair $(1,1)$ and plugged its values ($x=1, y=1$) into the equation $y = 4x - 5$. We performed the calculation: $1 = 4(1) - 5$, which simplified to $1 = 4 - 5$, and further to $1 = -1$. Since $1$ is definitely not equal to $-1$, the statement is false. Therefore, the ordered pair $(1,1)$ is not a solution to the equation $y = 4x - 5$. The answer is False.

Keep this method in your back pocket, guys. It’s a simple yet incredibly effective way to check your work and ensure you're on the right track. Happy problem-solving!