Resolvendo Sistemas De Equações: Encontre X E Y
Hey guys! Today, let's dive into the world of solving systems of equations. We're going to break down how to find the values of x and y in a simple system. This is super useful, whether you're brushing up on your math skills or just curious about how these equations work. We'll start with the basics, solve the system, and then see which answer options are correct. So, grab your pencils (or your favorite note-taking app), and let's get started. Get ready to flex those math muscles – it's going to be a fun ride!
Entendendo o Sistema de Equações
Alright, so what exactly is a system of equations? Basically, it's a set of two or more equations that we need to solve together. The goal? To find the values of the variables (in our case, x and y) that satisfy all the equations in the system. Think of it like a puzzle where the solution fits perfectly into each piece. In our example, we've got a simple system with two equations. Each equation represents a line, and the solution to the system is the point where those lines intersect on a graph. Getting the hang of it can really change the game when it comes to tackling more complex problems later on. Being able to visualize what we're working with helps us stay focused and on track with the material.
Here's the system we'll be working with:
- Equation 1: x + 2 = -7
- Equation 2: -5 + 6y = 3
Our mission is to find the values of x and y that make both of these equations true at the same time. Let's break it down step by step to find those secret numbers. Remember, each equation is like a little world of its own, but we want to find the common ground where everything aligns perfectly. The process we follow today will give us the base needed to tackle more complex system setups in the future. We'll go through this in detail to make sure we don't leave any room for error in our calculations. Understanding the equations can make your math journey a whole lot easier!
Resolvendo para x
Let's start by solving the first equation for x. It's pretty straightforward, but let's walk through it together to make sure we're on the same page. The first equation, x + 2 = -7, tells us that if we add 2 to the value of x, we get -7. To find x, we need to do the opposite of adding 2, which is subtracting 2. We'll subtract 2 from both sides of the equation to keep things balanced.
So, we have:
x + 2 - 2 = -7 - 2
Simplifying this, we get:
x = -9
That's it! We've found the value of x. It's -9. Now, let's move on to find the value of y.
Now, let's add a bit of detail to it. Think of the equation as a balance scale. Everything on one side must equal everything on the other side. When we change one side, we have to change the other to maintain this balance. This concept is fundamental in algebra, so understanding it here will benefit you as you solve more complex equations. By making sure we're following this simple rule, we can solve for any variable in a linear equation.
Resolvendo para y
Now, let's tackle the second equation: -5 + 6y = 3. Our goal here is to isolate y. To do this, we'll first get rid of the -5. The opposite of subtracting 5 is adding 5, so we add 5 to both sides of the equation:
-5 + 6y + 5 = 3 + 5
This simplifies to:
6y = 8
Now, to find y, we need to divide both sides by 6:
6y / 6 = 8 / 6
This gives us:
y = 8/6
We can simplify the fraction 8/6 by dividing both the numerator and the denominator by 2. This gives us:
y = 4/3
So, the value of y is 4/3. Now, we've found both x and y. We're on our way to selecting the correct answer.
Keep in mind that when we're solving for a variable, we're essentially rearranging the equation to get that variable by itself on one side. The key is to do the same thing to both sides to maintain the equation's balance. This technique will be useful as you get into solving more complex algebraic problems. By applying these methods, you'll be able to solve these with confidence, making your problem-solving skills shine.
Verificando as Alternativas
Now that we've found the values of x and y, let's check the given answer options to see which one matches our solution. We found that x = -9 and y = 4/3. Let's look at the options:
- x = -9 and y = 1
- x = -9 and y = 0
- x = -5 and y = 4/3
By comparing our results with the options, we can easily identify the correct answer. The process of checking the options helps us confirm that we've correctly solved the equations. It's a great way to double-check our work and gain confidence in our skills. Let's make sure we're on the right track before we declare victory. This step is about accuracy and precision, making sure everything aligns perfectly.
Identificando a Alternativa Correta
Okay, let's analyze the options we have and see which one lines up with our solutions. We know that x = -9 and y = 4/3. Remember that y = 4/3 is approximately 1.33. Now, let's look at the answer choices.
Option 1 says x = -9 and y = 1. This is close, but y isn't exactly 1, so it's not the correct answer.
Option 2 says x = -9 and y = 0. This is incorrect since we know that y is definitely not zero.
We did not have option 3 originally. But now, we'll imagine what the correct option could have been: x = -9 and y = 4/3. This matches our calculated values perfectly!
So, based on our calculations, the correct option should state x = -9 and y = 4/3. This demonstrates how to solve a system of equations, and then check against given choices. Understanding each step ensures you are confident when you have to solve problems on your own.
Conclusão
Great job, everyone! We successfully solved a system of equations, finding the values of x and y. This is a fundamental skill in math, and you've now practiced it. Remember to always double-check your work and to understand each step you take. Keep practicing, and you'll become a pro at solving these types of problems. Each system you solve gets easier and the process becomes second nature.
So, next time you come across a system of equations, you'll know exactly what to do. Keep up the amazing work, and don't hesitate to practice! Now you are ready to apply these skills to solve other problems, and you'll do great. Keep up the hard work, and good luck with your future math endeavors!