Solving For X And Y: 2x-y=4 & X+2y=3
Hey guys! Today, we're diving into a fundamental concept in algebra: solving simultaneous equations. Specifically, we're going to tackle the system:
- 2x - y = 4
- x + 2y = 3
This is a classic problem, and mastering it will give you a solid foundation for more complex mathematical challenges. So, grab your pencils, and let's get started!
Understanding the Problem
Before we jump into the solution, let's take a moment to understand what we're trying to achieve. The core goal is to find the values of x and y that make both equations true simultaneously. Think of it like finding the point where two lines intersect on a graph – that point's coordinates (x, y) will satisfy both equations.
We have two equations and two unknowns, which means we can use a couple of different methods to solve this. We'll explore two popular techniques: the substitution method and the elimination method. Both methods are effective, and choosing one often comes down to personal preference or which one seems more straightforward for a particular problem.
Why are Simultaneous Equations Important?
Now, you might be wondering, "Why should I care about solving these equations?" Well, simultaneous equations pop up everywhere in the real world! They're used in various fields like:
- Engineering: Calculating forces and stresses in structures.
- Economics: Modeling supply and demand.
- Computer science: Developing algorithms and solving optimization problems.
- Everyday life: Even something as simple as figuring out the cost of individual items when you have a combined price can involve simultaneous equations!
So, understanding this concept is definitely a worthwhile investment.
Method 1: The Substitution Method
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This effectively reduces the problem to a single equation with one unknown, which we can then solve easily.
Let's walk through the steps:
Step 1: Solve one equation for one variable
Look at our two equations:
- 2x - y = 4
- x + 2y = 3
Equation (2) looks easier to solve for x. Let's isolate x:
x = 3 - 2y
We've now expressed x in terms of y. This is our key substitution!
Step 2: Substitute into the other equation
Now, we'll substitute this expression for x into equation (1):
2(3 - 2y) - y = 4
Notice that we've replaced x with (3 - 2y). We now have a single equation with only y as the unknown.
Step 3: Solve for the remaining variable
Let's simplify and solve for y:
6 - 4y - y = 4
6 - 5y = 4
-5y = -2
y = 2/5
Great! We've found the value of y.
Step 4: Substitute back to find the other variable
Now that we know y = 2/5, we can substitute this value back into either equation (1) or (2) to find x. However, it's often easiest to substitute it back into the equation we used in step 1 (x = 3 - 2y):
x = 3 - 2(2/5)
x = 3 - 4/5
x = 11/5
So, we've found that x = 11/5.
Step 5: Verify your solution
It's always a good idea to check your solution by plugging the values of x and y back into the original equations to make sure they hold true. Let's do that:
- Equation (1): 2(11/5) - (2/5) = 22/5 - 2/5 = 20/5 = 4 (Correct!)
- Equation (2): (11/5) + 2(2/5) = 11/5 + 4/5 = 15/5 = 3 (Correct!)
Since our solution satisfies both equations, we've successfully found the values of x and y using the substitution method.
Method 2: The Elimination Method
The elimination method involves manipulating the equations so that the coefficients of one variable are opposites. When we add the equations together, that variable will be eliminated, leaving us with a single equation with one unknown.
Let's go through the steps using the same system of equations:
- 2x - y = 4
- x + 2y = 3
Step 1: Multiply equations to make coefficients opposites
We want to eliminate either x or y. Let's choose to eliminate y. Notice that the coefficient of y in equation (1) is -1, and in equation (2) it's 2. To make them opposites, we can multiply equation (1) by 2:
2 * (2x - y) = 2 * 4 => 4x - 2y = 8
Now we have the following system:
- 4x - 2y = 8
- x + 2y = 3
Notice that the coefficients of y are now -2 and 2, which are opposites!
Step 2: Add the equations
Now, we add the two equations together:
(4x - 2y) + (x + 2y) = 8 + 3
5x = 11
The y terms have been eliminated, leaving us with a simple equation in x.
Step 3: Solve for the remaining variable
Solve for x:
x = 11/5
We've found the value of x!
Step 4: Substitute back to find the other variable
Substitute x = 11/5 back into either of the original equations to solve for y. Let's use equation (2):
(11/5) + 2y = 3
2y = 3 - 11/5
2y = 4/5
y = 2/5
We've found that y = 2/5.
Step 5: Verify your solution
Just like before, let's check our solution:
- Equation (1): 2(11/5) - (2/5) = 22/5 - 2/5 = 20/5 = 4 (Correct!)
- Equation (2): (11/5) + 2(2/5) = 11/5 + 4/5 = 15/5 = 3 (Correct!)
The elimination method also gives us the solution x = 11/5 and y = 2/5.
The Solution
Using both the substitution and elimination methods, we've arrived at the same solution:
- x = 11/5
- y = 2/5
These values satisfy both equations in the system.
Tips and Tricks for Solving Simultaneous Equations
- Choose the easiest method: Sometimes, one method is clearly easier than the other. Look for opportunities to simplify the process.
- Be careful with signs: A common mistake is to make errors with negative signs. Double-check your work!
- Organize your work: Keep your steps clear and organized to avoid confusion.
- Practice makes perfect: The more you practice, the more comfortable you'll become with solving simultaneous equations.
Conclusion
Solving simultaneous equations is a valuable skill in mathematics and beyond. We've explored two powerful methods – substitution and elimination – and applied them to a specific example. Remember to understand the underlying concepts, practice regularly, and don't be afraid to try different approaches.
I hope this guide has been helpful, guys! Keep practicing, and you'll become a simultaneous equation pro in no time!