Solving 2x - Y = 9 And 4x + 2Y = 38 Find X And Y
Hey guys! Today, we're diving into the world of mathematics to tackle a classic problem: solving a system of linear equations. We've got two equations here: 2x - Y = 9 and 4x + 2Y = 38. Our mission is to find the values of x and y that satisfy both equations simultaneously. This is a fundamental concept in algebra, and mastering it opens the door to solving more complex problems in various fields, from engineering to economics. So, let's break it down step by step and make sure we understand each method thoroughly.
Why Solve Systems of Equations?
Before we jump into the solutions, let's quickly chat about why solving systems of equations is so important. Imagine you're trying to figure out the cost of two different items at a store, but you only have information about the total cost of combinations of these items. This is where systems of equations come to the rescue! They allow us to model real-world situations with multiple unknowns and find the values of those unknowns. Think about it: in fields like economics, you might need to determine equilibrium prices and quantities; in engineering, you might need to calculate forces acting on a structure; and in computer science, you might need to solve for variables in an algorithm. The possibilities are endless! So, by learning how to solve these equations, we're equipping ourselves with a powerful tool for tackling a wide range of problems.
Now, let's get back to our specific problem. We've got two methods to explore: solving for x and solving for y. We'll start with the method that involves isolating x.
Method 1: Solving for x
Okay, so the first method we're going to explore focuses on finding the value of x. There are a couple of ways we can approach this, but the most common is using either the substitution or elimination method. Let's start with the substitution method, which involves solving one equation for one variable and then substituting that expression into the other equation. This method is super handy when one of the equations is already solved for a variable or can be easily manipulated to do so.
The Substitution Method for x
Looking at our equations (2x - Y = 9 and 4x + 2Y = 38), the first equation seems easier to solve for Y. So, let's do that! We can rewrite 2x - Y = 9 as Y = 2x - 9. See how we just isolated Y? Now, we're going to take this expression for Y and substitute it into the second equation (4x + 2Y = 38). This is the key step in the substitution method. By replacing Y with (2x - 9), we'll have an equation with only one variable, x, which we can then solve.
So, substituting Y = 2x - 9 into 4x + 2Y = 38, we get:
4x + 2(2x - 9) = 38
Now, it's just a matter of simplifying and solving for x. First, we distribute the 2:
4x + 4x - 18 = 38
Next, we combine like terms:
8x - 18 = 38
Then, we add 18 to both sides:
8x = 56
And finally, we divide both sides by 8:
x = 7
Boom! We've found the value of x. It's 7. Now that we know x, we can plug it back into either of our original equations to solve for Y. Let's use the simpler one, 2x - Y = 9.
Substituting x = 7 into 2x - Y = 9, we get:
2(7) - Y = 9
14 - Y = 9
Subtracting 14 from both sides:
-Y = -5
And multiplying both sides by -1:
Y = 5
So, we've found that x = 7 and Y = 5. We've successfully solved the system of equations using the substitution method!
The Elimination Method for x
Now, let's explore the elimination method, which is another powerful technique for solving systems of equations. This method involves manipulating the equations so that either the x or Y coefficients are opposites. When we add the equations together, one of the variables will be eliminated, leaving us with a single equation in one variable. This method is particularly useful when the coefficients of one of the variables are already close to being opposites or multiples of each other.
Looking at our equations (2x - Y = 9 and 4x + 2Y = 38), we can see that the coefficients of x are 2 and 4. To eliminate x, we can multiply the first equation by -2. This will give us a -4x term, which is the opposite of the 4x term in the second equation.
Multiplying the first equation (2x - Y = 9) by -2, we get:
-4x + 2Y = -18
Now, we have the following system of equations:
-4x + 2Y = -18
4x + 2Y = 38
Next, we add the two equations together. Notice that the x terms cancel out:
(-4x + 4x) + (2Y + 2Y) = -18 + 38
0 + 4Y = 20
4Y = 20
Now, we divide both sides by 4:
Y = 5
We've found that Y = 5 using the elimination method. Now, we can substitute this value back into either of our original equations to solve for x. Let's use the first equation, 2x - Y = 9.
Substituting Y = 5 into 2x - Y = 9, we get:
2x - 5 = 9
Adding 5 to both sides:
2x = 14
And dividing both sides by 2:
x = 7
So, using the elimination method, we've also found that x = 7 and Y = 5. Both the substitution and elimination methods give us the same solution, which is a great way to check our work!
Method 2: Solving for Y
Alright, let's switch gears and focus on finding the value of Y first. Just like with solving for x, we can use either the substitution or elimination method. This time, we'll rearrange our steps to prioritize isolating Y. This is a fantastic way to show how flexible these methods are and how we can adapt them to suit the specific problem we're facing.
The Substitution Method for Y
Let's revisit our equations: 2x - Y = 9 and 4x + 2Y = 38. This time, instead of solving the first equation for Y directly, let's solve it for x. This might seem a bit counterintuitive since we're trying to find Y, but trust me, it'll work out! Solving for x gives us another way to approach the problem and reinforces our understanding of the substitution method.
So, let's solve 2x - Y = 9 for x. First, we add Y to both sides:
2x = Y + 9
Then, we divide both sides by 2:
x = (Y + 9) / 2
Now, we have an expression for x in terms of Y. We can substitute this into the second equation (4x + 2Y = 38).
Substituting x = (Y + 9) / 2 into 4x + 2Y = 38, we get:
4((Y + 9) / 2) + 2Y = 38
Simplifying, we have:
2(Y + 9) + 2Y = 38
Distributing the 2:
2Y + 18 + 2Y = 38
Combining like terms:
4Y + 18 = 38
Subtracting 18 from both sides:
4Y = 20
And dividing both sides by 4:
Y = 5
There we go! We've found Y = 5 using the substitution method, but this time by solving for x first. This illustrates that we have choices in how we apply the method, and sometimes one approach might be more convenient than another. Now that we know Y, we can plug it back into either of our original equations to solve for x. Let's use the first equation, 2x - Y = 9.
Substituting Y = 5 into 2x - Y = 9, we get:
2x - 5 = 9
Adding 5 to both sides:
2x = 14
And dividing both sides by 2:
x = 7
So, we've confirmed that x = 7 and Y = 5, even when we solve for Y first using the substitution method.
The Elimination Method for Y
Finally, let's tackle solving for Y using the elimination method. This will give us a complete picture of how to approach these types of problems and solidify our understanding of the elimination technique.
Looking at our equations (2x - Y = 9 and 4x + 2Y = 38), we can see that the coefficients of Y are -1 and 2. To eliminate Y, we can multiply the first equation by 2. This will give us a -2Y term, which is the opposite of the 2Y term in the second equation.
Multiplying the first equation (2x - Y = 9) by 2, we get:
4x - 2Y = 18
Now, we have the following system of equations:
4x - 2Y = 18
4x + 2Y = 38
Next, we add the two equations together. Notice that the Y terms cancel out:
(4x + 4x) + (-2Y + 2Y) = 18 + 38
8x + 0 = 56
8x = 56
Now, we divide both sides by 8:
x = 7
We've found that x = 7 using the elimination method. Now, we can substitute this value back into either of our original equations to solve for Y. Let's use the first equation, 2x - Y = 9.
Substituting x = 7 into 2x - Y = 9, we get:
2(7) - Y = 9
14 - Y = 9
Subtracting 14 from both sides:
-Y = -5
And multiplying both sides by -1:
Y = 5
So, using the elimination method, we've also found that x = 7 and Y = 5. We've now solved the system of equations using both the substitution and elimination methods, prioritizing both x and Y. This comprehensive approach demonstrates the power and flexibility of these techniques.
Solution and Verification
Alright, guys, after all that work, let's recap! We've found that x = 7 and Y = 5 is the solution to the system of equations 2x - Y = 9 and 4x + 2Y = 38. But before we celebrate, let's make absolutely sure our solution is correct. The best way to do this is to plug our values for x and Y back into the original equations and see if they hold true. This is a crucial step in problem-solving – always verify your solutions!
Let's start with the first equation, 2x - Y = 9. Substituting x = 7 and Y = 5, we get:
2(7) - 5 = 9
14 - 5 = 9
9 = 9
Awesome! The first equation checks out. Now, let's try the second equation, 4x + 2Y = 38. Substituting x = 7 and Y = 5, we get:
4(7) + 2(5) = 38
28 + 10 = 38
38 = 38
Fantastic! The second equation also holds true. Since our values for x and Y satisfy both equations, we can confidently say that our solution is correct. We've successfully navigated this system of equations and emerged victorious!
Conclusion
So, there you have it! We've thoroughly explored how to solve the system of equations 2x - Y = 9 and 4x + 2Y = 38 using both the substitution and elimination methods, focusing on solving for both x and Y. We've seen how these methods work, why they're valuable, and how to apply them effectively. Remember, the key to mastering these techniques is practice. The more you work with systems of equations, the more comfortable and confident you'll become in solving them.
Solving systems of equations is a fundamental skill in mathematics, and it's one that will serve you well in many areas of life. From calculating the best deals at the store to tackling complex problems in science and engineering, the ability to solve for unknowns is a powerful asset. So, keep practicing, keep exploring, and keep challenging yourself. You've got this!
If you have any questions or want to explore more examples, feel free to ask. Happy problem-solving, guys!
