Solving For X: A System Of Linear Equations

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Hey guys! Today, we're diving into a fun little problem from the world of mathematics. We're going to figure out how to find the value of 'x' when we have a system of linear equations. Don't worry, it's not as scary as it sounds! We'll break it down step by step so it's super easy to follow. So, grab your pencils, and let's get started!

Understanding the Problem

So, the burning question is: What is the value of 'x' in the solution to the system of linear equations? We're given two equations:

  1. y = 3x + 2
  2. y = x - 4

Our mission, should we choose to accept it (and we do!), is to find the value of 'x' that makes both of these equations true at the same time. Basically, we're looking for the point where these two lines intersect if we were to graph them. There are a few ways to tackle this, but we're going to use the substitution method because it's pretty straightforward. Understanding the problem is the first leap; solving it is the natural next step.

Solving the System of Equations

Alright, let's roll up our sleeves and get to the fun part: solving for 'x'. Since both equations are already solved for y, we can set them equal to each other. This is the heart of the substitution method. If y = 3x + 2 and y = x - 4, then it must be true that 3x + 2 = x - 4. See how we just replaced one y with the expression from the other equation? This is what simplifies the problem so that we only have one unknown variable.

Now we have a single equation with just 'x' in it. Let's get all the 'x' terms on one side and the constants on the other. Subtract x from both sides of the equation 3x + 2 = x - 4 to get 2x + 2 = -4. Next, we need to isolate the term with 'x'. Subtract 2 from both sides of the equation 2x + 2 = -4, and you will get 2x = -6. Lastly, to solve for 'x', divide both sides of the equation 2x = -6 by 2. This gives us x = -3. And there you have it! We've found the value of 'x'. But let's not stop here; we should always check our work to make sure we didn't make any silly mistakes.

Checking Our Work

Now, the key to being a math whiz is to always check your work. Seriously, it can save you from making simple mistakes. Plug x = -3 back into both original equations to see if they hold true.

For the first equation, y = 3x + 2, we substitute x = -3 to get y = 3(-3) + 2 = -9 + 2 = -7. So, y = -7.

For the second equation, y = x - 4, we substitute x = -3 to get y = -3 - 4 = -7. Again, y = -7.

Since we got the same y value from both equations when we plugged in x = -3, our solution is correct! This gives us confidence that we have solved the system correctly. Verifying the solution is just as important as solving the problem itself, as it ensures accuracy and understanding.

The Answer

So, after all that brain-busting work, what's the final answer? The value of 'x' in the solution to the system of linear equations is -3. That corresponds to answer choice B. Wasn't that fun? (Okay, maybe not fun for everyone, but hopefully, it was at least clear and understandable!)

Additional Tips for Solving Systems of Equations

Here are some extra tips and tricks to help you master solving systems of equations:

  • Choose the Right Method: Sometimes substitution is easier, and sometimes elimination is better. Look at the equations and see which method will require less work. If one of the variables is already isolated, substitution is usually the way to go.
  • Be Careful with Signs: One of the most common mistakes is messing up the signs when adding or subtracting equations. Double-check your work to make sure you haven't made any sign errors.
  • Write Neatly: This might sound silly, but writing neatly can help you avoid mistakes. When your work is organized, it's easier to see what you've done and spot any errors.
  • Practice, Practice, Practice: The more you practice, the better you'll get at solving systems of equations. Work through lots of problems, and don't be afraid to ask for help if you get stuck.

Remember, solving systems of equations is a fundamental skill in algebra, and it comes up in many different contexts. Mastering this skill will serve you well in future math courses and beyond. Keep practicing, and you'll become a pro in no time!

Real-World Applications

You might be wondering, "When am I ever going to use this stuff in real life?" Well, systems of equations pop up in all sorts of unexpected places. Here are a few examples:

  • Business: Businesses use systems of equations to model costs, revenue, and profit. For example, they might use a system of equations to determine the break-even point for a new product.
  • Engineering: Engineers use systems of equations to analyze circuits, design structures, and model fluid flow. These equations help them ensure that their designs are safe and efficient.
  • Economics: Economists use systems of equations to model supply and demand, analyze market equilibrium, and forecast economic trends. These models help them understand how different factors affect the economy.
  • Science: Scientists use systems of equations to model chemical reactions, analyze data, and make predictions. For example, they might use a system of equations to determine the rate of a chemical reaction.

So, the next time you're working on a system of equations, remember that you're learning a skill that has wide-ranging applications in the real world. Who knows, maybe one day you'll be using systems of equations to solve a problem that changes the world!

Conclusion

Alright, we've reached the end of our journey! We've successfully solved for 'x' in the given system of linear equations. Remember, the key is to understand the problem, choose the right method, and check your work. With a little practice, you'll be solving systems of equations like a pro.

Keep up the great work, and I'll see you next time with another fun math problem! Keep those brains sharp and those pencils moving. You've got this!