Hey guys! Today, we're diving into a fun little math problem: solving for x in a system of equations. Don't worry, it's not as intimidating as it sounds! We've got two equations here, and our mission, should we choose to accept it, is to figure out the numerical value that x represents. It's like a mathematical treasure hunt, and x marks the spot! So, grab your thinking caps, and let's get started!
The Equations at Hand
Before we plunge into the solution, let's take a good look at the equations we're dealing with. We have two equations, each with two variables, x and y. These variables are linked together in a way that, when we find the right values for x and y, both equations will be true. Think of it like a secret code where x and y are the keys. Here are our equations:
- x + 3y = 76
- x + y = 36
Now, the beauty of a system of equations is that we can use different methods to crack this code. There's the substitution method, the elimination method, and even graphical methods. But for this particular problem, the elimination method shines because it offers a super clean and straightforward way to isolate x. We'll walk through it step by step, so you can see exactly how it works.
Cracking the Code: The Elimination Method
Okay, let's talk strategy. The elimination method is all about getting rid of one variable so we can focus on the other. In our case, we want to find x, so we'll try to eliminate y first. The key to this method is to manipulate the equations so that the coefficients (the numbers in front of the variables) of one variable are the same, or more specifically, opposites. This way, when we add or subtract the equations, that variable will cancel out. In our problem, both equations already have x with a coefficient of 1, which is super convenient! But the coefficients of y are different (3 and 1).
So, here's the plan: We're going to subtract the second equation from the first. Why subtract? Because both equations have a positive x. Subtracting will eliminate x directly. If one x was positive and the other negative, we'd add the equations instead. Remember, the goal is to make one variable disappear temporarily, so we can solve for the other. Let's write it out:
( x + 3y ) - ( x + y ) = 76 - 36
Now, we just need to carefully distribute the subtraction and simplify. Remember, subtracting a whole expression means we're subtracting each term inside the parentheses. Pay close attention to the signs; that's where mistakes can easily happen!
Step-by-Step Elimination
Let's break down the subtraction step by step. When we subtract the second equation from the first, we're essentially distributing a -1 to each term in the second equation. This looks like:
x + 3y - x - y = 76 - 36
Notice how the +x in the second equation becomes -x when we subtract. Now, we can combine like terms on the left side of the equation. The x and -x cancel each other out (that's the elimination magic!), and we're left with:
3y - y = 76 - 36
Simplifying further, we get:
2y = 40
Now we're in the home stretch for finding y! We've reduced the equation to a simple one-variable problem. All that's left is to isolate y. How do we do that? By dividing both sides of the equation by the coefficient of y, which is 2. This will undo the multiplication and leave y all by itself.
Isolating and Solving for y
Alright, we're almost there! We have the equation 2y = 40. To isolate y, we need to get rid of that 2 that's multiplying it. The way we do that is by performing the opposite operation: division. We'll divide both sides of the equation by 2. It's crucial to do it on both sides to keep the equation balanced – think of it like a seesaw; whatever you do on one side, you need to do on the other to keep it level.
So, we divide both sides by 2:
(2y) / 2 = 40 / 2
On the left side, the 2s cancel out, leaving us with just y. On the right side, 40 divided by 2 is 20. So we have:
y = 20
Woohoo! We've found the value of y! This is a huge step. But remember, our ultimate goal is to find x. Now that we know y, we can use it like a cheat code to unlock the value of x. We can plug this value of y back into either of our original equations. It doesn't matter which one we choose; both will lead us to the correct value of x. Let's pick the simpler equation to make our lives easier.
Plugging y Back In: Finding x
Now that we know y = 20, we can substitute this value into one of our original equations to solve for x. Remember those equations? They were:
- x + 3y = 76
- x + y = 36
Let's go with the second equation, x + y = 36, because it looks a little simpler. We'll replace y with 20:
x + 20 = 36
Now we have a simple equation with only x as the unknown. To isolate x, we need to get rid of that +20. How do we do that? By subtracting 20 from both sides of the equation. This is the opposite of addition, and it will undo the addition, leaving x all by itself.
Subtracting 20 from both sides, we get:
x + 20 - 20 = 36 - 20
Simplifying, we have:
x = 16
And there we have it! We've solved for x. It's like finding the final piece of the puzzle. Now we know both x and y. But before we declare victory, let's do one last crucial step: verification.
The Grand Finale: Verification
Okay, we've found x = 16 and y = 20. But math, like life, sometimes throws curveballs. It's always a good idea to double-check our work, especially in a system of equations. We need to make sure that these values actually work in both of our original equations. This is like testing our secret code in both locks to make sure it opens them.
Let's plug x = 16 and y = 20 into our first equation, x + 3y = 76:
16 + 3(20) = 76
Simplifying, we get:
16 + 60 = 76
76 = 76
It checks out! The equation holds true. Now, let's do the same for our second equation, x + y = 36:
16 + 20 = 36
36 = 36
It checks out again! Both equations are satisfied by our values of x and y. This means we've successfully cracked the code! We can confidently say that x = 16 is the solution.
Conclusion: We Did It!
Guys, we did it! We successfully navigated the world of systems of equations and found the value of x. We used the elimination method, which is a powerful tool in algebra, and we even verified our answer to make sure it was spot on. Remember, the key to solving these problems is to break them down into smaller, manageable steps. Don't be afraid to try different approaches, and always double-check your work. Math can be a fun adventure when you approach it with the right mindset. So, keep practicing, keep exploring, and keep those mathematical gears turning!
