Solving 2x - Y = -2 And X + 2y = 4 A Step-by-Step Guide
Hey guys! Today, we're diving into a classic math problem: solving a system of linear equations. Specifically, we'll be tackling the system:
- 2x - y = -2
- x + 2y = 4
This is a fundamental concept in algebra, and mastering it will open doors to more advanced topics. So, let's break it down step-by-step and make sure you've got a solid understanding. We will explore different methods to tackle this, ensuring you're well-equipped to handle similar problems in the future. Get ready to put on your math hats – let's get started!
Understanding Systems of Equations
Before we jump into the solution, let's quickly recap what a system of equations actually is. Essentially, it's a set of two or more equations containing the same variables. Our goal is to find the values of these variables that satisfy all equations in the system simultaneously. In simpler terms, we're looking for the point where the lines represented by these equations intersect on a graph. Think of it like a puzzle where each equation is a clue, and we need to piece them together to find the hidden solution.
In our case, we have two linear equations, meaning they represent straight lines when graphed. The solution to the system will be the single point (x, y) where these two lines cross each other. But how do we find this point without actually drawing the graph? That's where algebraic methods come in handy, which we will explore in detail below. Understanding the concept of a system of equations is crucial because it appears in various real-world applications, from calculating mixtures in chemistry to determining optimal production levels in economics. So, pay close attention, and let's unlock the secrets of solving these equations!
Method 1: Substitution Method
The substitution method is a powerful technique for solving systems of equations. The core idea is to isolate one variable in one equation and then substitute that expression into the other equation. This effectively reduces the system to a single equation with a single variable, which we can then solve easily. Let's see how this works with our example:
Step 1: Isolate a Variable
Look at our two equations:
- 2x - y = -2
- x + 2y = 4
It seems easier to isolate 'x' in the second equation. So, let's rearrange it:
x = 4 - 2y
Now we have an expression for 'x' in terms of 'y'. This is our key to substitution.
Step 2: Substitute
Now, we'll substitute this expression for 'x' (which is 4 - 2y) into the first equation:
2(4 - 2y) - y = -2
See what we did? We replaced 'x' in the first equation with its equivalent expression in terms of 'y'. This gives us a new equation that only contains the variable 'y'.
Step 3: Solve for y
Let's simplify and solve this equation for 'y':
8 - 4y - y = -2
Combine the 'y' terms:
8 - 5y = -2
Subtract 8 from both sides:
-5y = -10
Divide both sides by -5:
y = 2
Great! We've found the value of 'y'.
Step 4: Solve for x
Now that we know y = 2, we can plug this value back into either of the original equations to solve for 'x'. However, it's usually easiest to use the equation where we already isolated 'x':
x = 4 - 2y
Substitute y = 2:
x = 4 - 2(2)
x = 4 - 4
x = 0
Step 5: The Solution
We've found x = 0 and y = 2. So, the solution to the system of equations is the ordered pair (0, 2). This means that the lines represented by these two equations intersect at the point (0, 2) on a graph.
The substitution method, guys, is all about strategic replacement. By isolating a variable and substituting its expression, we simplify the problem into a single-variable equation. Remember to double-check your work and ensure your solution satisfies both original equations. Now, let's move on to another powerful technique: the elimination method.
Method 2: Elimination Method
The elimination method, also known as the addition method, is another fantastic technique for solving systems of equations. Unlike substitution, this method focuses on eliminating one of the variables by strategically adding or subtracting the equations. This is achieved by manipulating the equations so that the coefficients of one variable are opposites. Let's see how it works with our system:
- 2x - y = -2
- x + 2y = 4
Step 1: Align and Prepare
Our goal is to make the coefficients of either 'x' or 'y' opposites. Looking at our equations, it seems easier to target the 'y' variable. Notice that the coefficient of 'y' in the first equation is -1, and in the second equation, it's 2. To make them opposites, we can multiply the first equation by 2:
2 * (2x - y) = 2 * (-2)
This gives us:
4x - 2y = -4
Now we have the following system:
- 4x - 2y = -4
- x + 2y = 4
See how the coefficients of 'y' are now -2 and 2? They're opposites!
Step 2: Eliminate
Now comes the magic! We add the two equations together. When we do this, the 'y' terms will cancel each other out:
(4x - 2y) + (x + 2y) = -4 + 4
This simplifies to:
5x = 0
Step 3: Solve for x
Now we have a simple equation to solve for 'x':
5x = 0
Divide both sides by 5:
x = 0
We've found the value of 'x'!
Step 4: Solve for y
Now that we know x = 0, we can plug this value back into either of the original equations to solve for 'y'. Let's use the second equation:
x + 2y = 4
Substitute x = 0:
0 + 2y = 4
2y = 4
Divide both sides by 2:
y = 2
Step 5: The Solution
Just like with the substitution method, we've found x = 0 and y = 2. So, the solution to the system of equations is the ordered pair (0, 2).
The elimination method, guys, shines when the equations are nicely aligned, and it's easy to create opposite coefficients. The key is to strategically multiply one or both equations to set up the elimination. Remember to add or subtract the equations carefully, ensuring you combine like terms correctly. Now that we've explored two powerful algebraic methods, let's recap the solution and talk about verifying our answer.
Verifying the Solution
It's always a good practice to verify your solution, guys. This helps you catch any potential errors and builds confidence in your answer. To verify our solution (0, 2), we simply plug these values back into the original equations and see if they hold true.
Equation 1: 2x - y = -2
Substitute x = 0 and y = 2:
2(0) - 2 = -2
0 - 2 = -2
-2 = -2 (This is true!)
Equation 2: x + 2y = 4
Substitute x = 0 and y = 2:
0 + 2(2) = 4
0 + 4 = 4
4 = 4 (This is also true!)
Since our solution (0, 2) satisfies both original equations, we can confidently say that it is the correct solution to the system. Verification, guys, is your safety net in math. It's a quick step that can save you from making careless mistakes. Make it a habit, and you'll become a more accurate and confident problem solver.
Conclusion
Alright guys, we've successfully solved the system of equations:
- 2x - y = -2
- x + 2y = 4
We explored two powerful methods: the substitution method and the elimination method. Both methods led us to the same solution: (0, 2). We also emphasized the importance of verifying your solution to ensure accuracy.
Solving systems of equations is a crucial skill in algebra and beyond. It lays the foundation for more complex mathematical concepts and has applications in various fields. So, keep practicing, guys, and master these techniques. Remember, the key is to understand the underlying concepts and choose the method that best suits the problem. Keep up the great work, and happy problem-solving!
