Solving 2x - Y = Y And X + 2y = 2: A Step-by-Step Guide
Hey guys! Ever stumbled upon a system of equations that looks like a tangled mess? Well, today we're going to untangle one such beast: 2x - y = y and x + 2y = 2. This isn't just about finding the values of x and y; it's about understanding the underlying concepts and different methods we can use to crack these problems. So, buckle up, and let's dive into the fascinating world of linear equations!
Understanding the Equations
Before we jump into solving, let's break down what these equations actually represent. The first equation, 2x - y = y, might look a bit odd at first glance. Notice that y appears on both sides of the equation. This is a crucial observation. We can simplify this equation by adding y to both sides, giving us 2x = 2y. Now it's starting to look a bit cleaner, right? We can further simplify this by dividing both sides by 2, leading us to a very elegant equation: x = y. This tells us that the value of x is exactly the same as the value of y. This is a powerful piece of information that we'll use later on.
The second equation, x + 2y = 2, is a more standard linear equation. It represents a straight line when graphed on a coordinate plane. The equation tells us that if we take the value of x and add twice the value of y, the result will always be 2. Think of it like a balancing act – x and y have to work together to maintain this balance.
Together, these two equations form a system of linear equations. A system of equations is simply a set of two or more equations that we are trying to solve simultaneously. In other words, we're looking for values of x and y that satisfy both equations at the same time. This point (x, y) is where the lines represented by the equations intersect on a graph. Solving a system of equations is like finding the meeting point of these lines.
Why is solving systems of equations important, you ask? Well, these systems pop up everywhere in real-world applications! From calculating the break-even point in business to determining the optimal mix of ingredients in a recipe, systems of equations are a fundamental tool in problem-solving. Understanding how to solve them opens doors to tackling a wide range of challenges.
Methods for Solving: Substitution
Okay, now that we understand the equations, let's get down to business and solve them! One of the most common and versatile methods for solving systems of equations is substitution. The substitution method involves solving one equation for one variable and then substituting 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 apply this method to our system. Remember that we simplified the first equation to x = y. This is perfect for substitution! We've already solved for x in terms of y. Now, we'll take this expression for x and substitute it into the second equation, x + 2y = 2. Replacing x with y gives us y + 2y = 2. See what we did there? We've eliminated x and now we have an equation with only y.
Now, let's simplify and solve for y. Combining the y terms, we get 3y = 2. To isolate y, we divide both sides of the equation by 3, which gives us y = 2/3. Fantastic! We've found the value of y. But we're not done yet; we still need to find the value of x.
This is where the substitution method really shines. We already know that x = y, and we now know that y = 2/3. Therefore, we can directly substitute the value of y into the equation x = y to find x. This gives us x = 2/3. So, we've found it! The solution to the system of equations is x = 2/3 and y = 2/3. This means that the point where the lines represented by these equations intersect is (2/3, 2/3).
Substitution is a powerful technique because it allows us to break down a complex problem into simpler steps. By isolating one variable and substituting its expression into another equation, we can systematically eliminate variables and solve for the unknowns. This method is particularly useful when one of the equations is already solved for one variable or can be easily rearranged to do so.
Methods for Solving: Elimination
Another powerful technique for solving systems of equations is the elimination method. This method focuses on eliminating one of the variables by manipulating the equations so that either the x coefficients or the y coefficients are opposites. When we add the equations together, one of the variables will cancel out, leaving us with a single equation with a single variable.
Let's see how this works with our system: 2x - y = y and x + 2y = 2. As before, let's simplify the first equation to x = y. Now we have our system as: x = y and x + 2y = 2. To use elimination, we need to arrange the equations so that the x or y terms line up. Let's rewrite the first equation as x - y = 0 (by subtracting y from both sides). Now our system looks like this:
- x - y = 0
- x + 2y = 2
Notice that the x coefficients are already the same (both are 1). To eliminate x, we can subtract the first equation from the second equation. This gives us:
(x + 2y) - (x - y) = 2 - 0
Simplifying the left side, we get x + 2y - x + y = 3y. Simplifying the right side, we get 2. So our new equation is 3y = 2. This is the same equation we got when using substitution! Solving for y, we divide both sides by 3, resulting in y = 2/3. Just as before, we've found the value of y.
Now, to find x, we can substitute the value of y back into either of the original equations. Let's use the simplified equation x = y. Substituting y = 2/3, we get x = 2/3. Again, we arrive at the same solution: x = 2/3 and y = 2/3.
The elimination method is particularly useful when the coefficients of one of the variables are easily made opposites by multiplying one or both equations by a constant. It provides a systematic way to eliminate variables and simplify the system, often leading to a straightforward solution. While in this case, substitution was arguably simpler due to the direct relationship x = y, the elimination method showcases a different approach that can be advantageous in other scenarios.
Verifying the Solution
It's always a good idea to verify our solution to make sure we haven't made any mistakes along the way. This is especially important in mathematics, as a small error can throw off the entire result. To verify our solution, we simply substitute the values we found for x and y back into the original equations and check if they hold true.
Our solution is x = 2/3 and y = 2/3. Let's plug these values into the first original equation, 2x - y = y:
2(2/3) - (2/3) = (2/3)
Simplifying the left side, we get 4/3 - 2/3 = 2/3. The equation holds true! Now, let's check the second original equation, x + 2y = 2:
(2/3) + 2(2/3) = 2
Simplifying the left side, we get 2/3 + 4/3 = 6/3, which simplifies to 2. Again, the equation holds true! Since our solution satisfies both original equations, we can be confident that it is correct. This verification step is a crucial part of the problem-solving process, helping us to avoid errors and build confidence in our results.
Graphical Interpretation
Beyond the algebraic methods of substitution and elimination, we can also understand systems of equations graphically. Each linear equation in a system represents a straight line on a coordinate plane. The solution to the system is the point where the lines intersect. This intersection point represents the values of x and y that satisfy both equations simultaneously.
Let's visualize our system: 2x - y = y and x + 2y = 2. We already simplified the first equation to x = y. This equation represents a line that passes through the origin (0, 0) and has a slope of 1. In other words, for every unit we move to the right along the x-axis, we also move one unit up along the y-axis. The line forms a 45-degree angle with both axes.
The second equation, x + 2y = 2, represents a different line. To graph this line, we can find two points that lie on it. For example, if we set x = 0, we get 2y = 2, so y = 1. This gives us the point (0, 1). If we set y = 0, we get x = 2, giving us the point (2, 0). We can then draw a line through these two points.
If you were to graph these two lines, you would see that they intersect at the point (2/3, 2/3). This is the same solution we found using substitution and elimination! The graphical representation provides a visual confirmation of our algebraic solution. It also helps to solidify our understanding of what a solution to a system of equations actually means – it's the point where the lines representing the equations cross paths.
Real-World Applications
So, we've mastered solving the system 2x - y = y and x + 2y = 2. But where does this knowledge come in handy in the real world? Systems of equations are surprisingly versatile and show up in a wide range of applications.
Finance: Imagine you're comparing two investment options. One option offers a fixed interest rate, while the other has a variable rate plus a bonus. You can set up a system of equations to model the returns from each investment and determine when one option becomes more profitable than the other. The point of intersection of the lines representing these equations would show the investment amount and duration where the returns are equal.
Business: Businesses often use systems of equations to determine their break-even point. This is the point where total revenue equals total costs. By setting up equations for revenue and costs, a business can find the sales volume needed to cover all expenses. This helps in making crucial decisions about pricing, production, and overall business strategy.
Science and Engineering: In physics, systems of equations are used to analyze forces, motion, and electrical circuits. In chemistry, they're used to balance chemical equations. Engineers use them in designing structures, analyzing stress and strain, and optimizing systems.
Everyday Life: Even in everyday situations, you might unconsciously use systems of equations. For example, when you're shopping and trying to decide between two deals, you might be setting up a mental system of equations to compare the total cost of each option. Or, if you're planning a trip and trying to figure out how much time to allocate to different activities, you might be using a system of equations to balance your schedule.
The ability to solve systems of equations is a valuable skill that can be applied to a multitude of problems in various fields. Understanding the different methods, like substitution and elimination, and the graphical interpretation allows us to tackle these problems with confidence and efficiency. So, next time you encounter a situation that seems complex, remember that a system of equations might just be the key to unlocking the solution!
Conclusion
Guys, we've really taken a deep dive into the system of equations 2x - y = y and x + 2y = 2. We explored different methods for solving it – substitution and elimination – and even visualized the solution graphically. We also saw how these skills translate into real-world applications, from finance and business to science and everyday decision-making. Hopefully, this breakdown has not only helped you understand how to solve this specific system but also given you a solid foundation for tackling other systems of equations in the future. Remember, practice makes perfect, so keep those equations coming! You've got this!
