Solving Equations: Find X And Y, Then Calculate X.Y
Hey math enthusiasts! Let's dive into a fun problem where we get to flex our equation-solving muscles. We're given a cool little setup: (2; 3x-y) = (x+y; 14). Our mission, should we choose to accept it, is to find the values of x and y, and then calculate their product, which is X.Y. Sounds like a blast, right?
This type of problem is all about understanding that when two ordered pairs are equal, their corresponding components must also be equal. Think of it like a treasure hunt; we have two clues, and each clue gives us a piece of the puzzle. The first clue tells us that the first elements of both pairs are equal, and the second clue tells us that the second elements are equal. So, we'll break this down step-by-step, making sure we don't miss a beat. We will use the concepts of algebra to solve this problem, specifically focusing on systems of equations. Understanding how to set up and solve these equations is a fundamental skill in algebra, and it's super useful for all sorts of real-world problems. By the end of this, you will be able to approach similar problems with confidence!
Alright, let's get started. Remember our equation: (2; 3x-y) = (x+y; 14). This means we have two separate equations hidden inside. Let's pull them out. First, we know that 2 = x + y, because the first elements of the ordered pairs are equal. This is our first equation! Cool, huh? Now, let's look at the second elements. We have 3x - y = 14. This is our second equation. We now have a system of two equations with two variables. It's time to put on our detective hats and solve this mystery! We are trying to find the value of x and y. So, we need to solve the system of equations. There are several methods we can use, such as substitution, elimination, or graphing, but for this problem, we'll try to use the substitution method to solve for x and y. The substitution method is great because it's usually straightforward, especially when one of your equations is already close to being solved for a variable. In order to solve for x and y, it's important to set up the system of equations correctly. So let's write them down: Equation 1: 2 = x + y, and Equation 2: 3x - y = 14. Now, the next step is to isolate one of the variables. Let's make it easy on ourselves and rearrange our first equation (2 = x + y) to solve for x. This gives us x = 2 - y. Now we have an equation for x in terms of y. This is where the magic of substitution happens. Since we know what x equals, we can plug that value into our second equation (3x - y = 14).
Step-by-Step Solution to Find x and y
Now, let's go through the steps of solving this problem. First, we'll deal with the equation: 2 = x + y. We're going to use this equation to help us find the values of x and y. Remember, the goal is to isolate one variable in terms of the other. Looking at our equation, the easiest variable to isolate seems to be x. To do this, we can subtract y from both sides of the equation. This gives us x = 2 - y. This equation tells us that x is equal to 2 minus y. Awesome! Now we know what x is in terms of y. Next, we're going to plug this expression for x into the second equation: 3x - y = 14. Instead of writing x, we're going to write (2 - y). So our equation becomes 3(2 - y) - y = 14. Now we get to solve for y! Let's simplify this equation. We start by distributing the 3 across the terms inside the parentheses: 3 * 2 = 6 and 3 * -y = -3y. So our equation is now 6 - 3y - y = 14. Combining like terms (the y terms), we have 6 - 4y = 14. The next step is to get the y term by itself. Let's subtract 6 from both sides of the equation. This gives us -4y = 8. Now we're close to finding the value of y. All we need to do is divide both sides of the equation by -4. So, y = 8 / -4, which simplifies to y = -2. Amazing, we've found the value of y! Now that we know y, we can easily find x by plugging y back into our equation x = 2 - y. If we substitute y = -2, we get x = 2 - (-2). Remember that subtracting a negative number is the same as adding a positive number. So, x = 2 + 2, which gives us x = 4. Awesome! We've found the values of both x and y: x = 4 and y = -2. We have successfully solved for the values of both variables, completing a significant step in solving the problem. We applied algebraic techniques such as substitution and simplification to find the value of x and y. Now, let's put it into practice and find the value of X.Y.
Now, let's take a closer look at the steps.
Step 1: Separate the Equations
From the original equation (2; 3x - y) = (x + y; 14), we can extract two separate equations. The first one comes from equating the first elements of the ordered pairs: 2 = x + y. The second equation is formed by equating the second elements: 3x - y = 14. These equations are the keys to unlock the values of x and y. Understanding how these equations relate to each other is crucial to solve for x and y. These steps are critical because they lay the foundation for the subsequent steps.
Step 2: Solve for One Variable (x or y) in One of the Equations
Let's take our first equation, 2 = x + y, and solve for x. To isolate x, we can rewrite the equation as x = 2 - y. This sets the stage for using substitution to find the value of x and y. We can now substitute the expression for x into the second equation to solve for y. This step is super important because it simplifies the system of equations. When we isolate a variable, we get a clear path to use substitution later on, which helps us to find the value of x and y. It's all connected, and this step is super crucial.
Step 3: Substitute and Solve for the Remaining Variable
Now that we have x = 2 - y, we'll substitute this into our second equation: 3x - y = 14. So, it becomes 3(2 - y) - y = 14. Now, let's simplify this. Distribute the 3: 6 - 3y - y = 14. Combine like terms: 6 - 4y = 14. Subtract 6 from both sides: -4y = 8. Divide by -4: y = -2. Boom! We've found the value of y. This step is a game-changer! When we substitute and simplify, we reduce the problem to a single variable equation, making it much easier to solve. The elimination of one variable and the simplification of the equation are key elements here. We just substitute, simplify, and solve for y. This process is very important to find the value of x and y.
Step 4: Substitute the Known Variable to Find the Other Variable
Now that we know y = -2, we can plug this value back into the equation x = 2 - y. So, x = 2 - (-2), which simplifies to x = 2 + 2 = 4. Voila! We've found that x = 4. This is a crucial step to solve this equation. It completes the process of finding the values of x and y. This step is simple but effective, we use the value of y we found to get the value of x.
Calculating the Product of X.Y
Now that we've found the values of x and y, it's time for the grand finale: calculating X.Y. Remember, we found that x = 4 and y = -2. The product X.Y simply means we multiply these two values together. So, X.Y = 4 * (-2). Multiplying these gives us X.Y = -8. And there you have it, folks! The answer to our problem is -8. Isn't math awesome?
So, to recap, we used the properties of equal ordered pairs to set up two equations. We then used substitution to solve for x and y. We found that x = 4 and y = -2. Finally, we calculated their product, which is -8. This means that the value of X.Y is -8. Awesome!
This entire process shows how we can use a combination of different techniques to solve this problem. These techniques can be used to solve many more problems.
Why This Matters
Understanding how to solve systems of equations is a fundamental skill in mathematics. It's not just about getting a number; it's about the process. This method is applicable in numerous real-world scenarios, from calculating the intersection of two lines to modeling complex systems. Being able to break down a problem, set up equations, and solve for unknowns is a powerful tool. It teaches you logical thinking, problem-solving, and critical analysis skills. This can be applied in many aspects of your life. This isn't just a math problem, it's about building a strong foundation in problem-solving. This knowledge can also be very useful to solve different kind of problems.
Conclusion: We Did It!
Alright, guys, we made it! We successfully found the values of x and y and then calculated their product. Remember, when you're faced with a problem like this, take it one step at a time. Break it down, use the right tools, and don't be afraid to double-check your work. Each step is a puzzle piece, and solving it is very rewarding. Youâre building your problem-solving skills, and that is awesome. Keep practicing, and you'll become a math whiz in no time. Congratulations, you are one step closer to mastering algebra! Feel free to practice on other similar problems to solidify your knowledge. Good luck! Keep up the great work! You've got this!