Solving For X And Y: A Step-by-Step Guide
Hey guys! Let's dive into a classic math problem where we need to figure out the values of two real numbers, X and Y. This is a super common type of question you'll see in algebra, and mastering it will really boost your problem-solving skills. We're given two equations: X + Y = 10 and X - Y = 2. Our mission, should we choose to accept it (and we totally do!), is to find out what X and Y are. We will explore different methods to solve this problem and clearly explain each step. Stick with me, and you'll be a pro at these in no time!
Understanding the Problem
Before we jump into solving, let's break down what we're dealing with. The heart of the matter lies in understanding that we have a system of two linear equations. Each equation represents a straight line, and the solution we're hunting for is the point where these lines intersect. Think of it as finding the exact spot where both equations agree. To kick things off, let's rewrite the equations to make them crystal clear:
- X + Y = 10
- X - Y = 2
Our goal is to find the values for X and Y that satisfy both equations simultaneously. We have a few cool methods to tackle this, but today, we'll focus on the elimination method and the substitution method. These are like the dynamic duo of algebra problem-solving!
Method 1: Elimination Method
The elimination method is like the ninja technique of algebra – it's all about making one of the variables disappear! The key here is to add or subtract the equations in a way that one of the variables cancels out. Looking at our equations, notice anything special? Yep, the Y terms have opposite signs (+Y and -Y). This is our golden ticket! When we add the two equations together, the Y terms will neatly eliminate each other. Let's do it:
(X + Y) + (X - Y) = 10 + 2
Now, let's simplify. Combine those like terms:
2X = 12
See how the Y vanished? Awesome! Now we have a simple equation with just X. To solve for X, we just divide both sides by 2:
X = 6
Boom! We found X. But hold on, we're not done yet – we still need Y. Now that we know X, we can plug it back into either of the original equations to solve for Y. Let's use the first equation, X + Y = 10. Substitute X = 6:
6 + Y = 10
To isolate Y, subtract 6 from both sides:
Y = 4
And there we have it! X = 6 and Y = 4. We've successfully navigated the elimination method. But let's not stop there; let's explore another equally powerful technique.
Method 2: Substitution Method
The substitution method is like being a master of disguise – we'll solve one equation for one variable and then substitute that expression into the other equation. This might sound a bit complicated, but trust me, it's super effective! Let's start with our original equations again:
- X + Y = 10
- X - Y = 2
We need to pick one equation and solve for one variable. Let's go with the first equation, X + Y = 10, and solve for X. To do this, subtract Y from both sides:
X = 10 - Y
Great! Now we have X expressed in terms of Y. This is our disguise. Next, we'll substitute this expression for X into the second equation, X - Y = 2:
(10 - Y) - Y = 2
See what we did? We replaced X with (10 - Y). Now we have an equation with only Y, which we can solve. Let's simplify:
10 - 2Y = 2
Subtract 10 from both sides:
-2Y = -8
Now, divide both sides by -2:
Y = 4
Woo-hoo! We found Y using the substitution method. Now, just like before, we can plug this value of Y back into either of the original equations to solve for X. Or, even easier, we can use the expression we already found, X = 10 - Y. Substitute Y = 4:
X = 10 - 4
X = 6
And just like that, we've confirmed our answer: X = 6 and Y = 4. We've conquered this problem using both the elimination method and the substitution method! You're becoming a true algebra whiz!
Verifying the Solution
Okay, we've got our answer, but let's be super sure it's correct. It's always a good idea to double-check your work, especially in math. To verify our solution, we'll plug X = 6 and Y = 4 back into both of the original equations:
- X + Y = 10 --> 6 + 4 = 10 (This checks out!)
- X - Y = 2 --> 6 - 4 = 2 (This checks out too!)
Since our values for X and Y satisfy both equations, we can confidently say that our solution is correct. Pat yourself on the back – you've nailed it!
Identifying the Correct Option
Now that we've solved for X and Y and verified our solution, let's take a look at the answer choices provided in the original question. We had four options:
A) X = 6 and Y = 4 B) X = 5 and Y = 5 C) X = 8 and Y = 2 D) X = 7 and Y = 3
Comparing our solution (X = 6, Y = 4) with the options, it's clear that option A is the correct answer. We've not only solved the problem but also identified the right choice! High five!
Why Other Options Are Incorrect
To really master this, let's quickly see why the other options are wrong. This helps solidify our understanding of the problem and the solution process. Remember, the correct values for X and Y must satisfy both equations. Let's test the other options:
- Option B) X = 5 and Y = 5
- Equation 1: 5 + 5 = 10 (Checks out)
- Equation 2: 5 - 5 = 0 (Does NOT equal 2)
- So, Option B is incorrect.
- Option C) X = 8 and Y = 2
- Equation 1: 8 + 2 = 10 (Checks out)
- Equation 2: 8 - 2 = 6 (Does NOT equal 2)
- So, Option C is incorrect.
- Option D) X = 7 and Y = 3
- Equation 1: 7 + 3 = 10 (Checks out)
- Equation 2: 7 - 3 = 4 (Does NOT equal 2)
- So, Option D is incorrect.
As you can see, only Option A satisfies both equations, making it the one and only correct answer. Understanding why the wrong answers are wrong is just as important as knowing why the right answer is right!
Key Takeaways
Alright, we've reached the end of our X and Y adventure! Let's recap the key things we've learned:
- System of Equations: We tackled a system of two linear equations, understanding that the solution is the point where the lines intersect.
- Elimination Method: We mastered the elimination method by adding the equations to eliminate one variable, making it easier to solve for the other.
- Substitution Method: We became substitution ninjas, solving one equation for a variable and substituting that expression into the other equation.
- Verification: We emphasized the importance of verifying our solution by plugging the values back into the original equations.
- Correct Option: We successfully identified the correct answer choice and understood why the other options were incorrect.
Practice Makes Perfect
Solving for variables like X and Y is a fundamental skill in algebra. The more you practice, the more confident you'll become. Try tackling similar problems, and don't be afraid to experiment with different methods. Math is like a puzzle, and you've got the tools to solve it!
So, keep practicing, keep exploring, and remember, math can be fun! You've got this!