Solving System Of Equations: X+y=5, X-2y=-1
Hey guys! Today, we're diving into a classic math problem: solving a system of equations. Specifically, we're going to tackle the system:
- x + y = 5
- x - 2y = -1
If you're scratching your head, don't worry! We'll break it down step-by-step so even if math isn't your favorite subject, you’ll get the hang of it. We'll explore different methods to solve this, ensuring you grasp the underlying concepts. This isn't just about getting the answer; it’s about understanding the process. Systems of equations pop up everywhere, from everyday budgeting to more complex scientific calculations. Mastering them now will set you up for success in future math endeavors and beyond. So, let's put on our thinking caps and get started!
Understanding Systems of Equations
Before we jump into solving, let's make sure we're all on the same page about what a system of equations actually is. Think of it as a set of two or more equations that share the same variables. In our case, we have two equations, and both use the variables 'x' and 'y'. The goal is to find the values for 'x' and 'y' that make both equations true at the same time. These values represent the point where the lines represented by the equations intersect on a graph. So, solving a system of equations is like finding the sweet spot where everything lines up perfectly.
There are a few common methods for solving these systems, and we'll be focusing on two today: substitution and elimination. Each method has its strengths, and the best one to use often depends on the specific equations you're dealing with. Think of it like having different tools in a toolbox – some are better suited for certain jobs than others. We'll walk through each method, highlighting the steps and the logic behind them. This way, you'll not only be able to solve this particular system but also have the skills to tackle similar problems in the future. Ready to explore these methods? Let’s jump in!
Method 1: Solving by Substitution
The substitution method is like a clever way of rearranging information to make the problem simpler. The basic idea is to solve one equation for one variable, and then substitute that expression into the other equation. This way, you'll end up with a single equation with only one variable, which is much easier to solve. Once you've found the value of that variable, you can plug it back into one of the original equations to find the value of the other variable.
Let's apply this to our system:
- x + y = 5
- x - 2y = -1
Step 1: Solve one equation for one variable.
Looking at the equations, it seems easiest to solve the first equation for 'x'. We can do this by subtracting 'y' from both sides:
x = 5 - y
Now we have an expression for 'x' in terms of 'y'. This is the key to the substitution method! This step is crucial because it isolates one variable, allowing us to express it in terms of the other. By doing so, we create a pathway to reduce the complexity of the system. The goal here is to find the most straightforward equation to manipulate, making the subsequent steps as clean as possible. Practice in identifying these opportunities will significantly improve your problem-solving efficiency. Remember, the choice of which variable to isolate can greatly impact the difficulty of the process, so choose wisely!
Step 2: Substitute the expression into the other equation.
Now, we'll substitute this expression for 'x' (which is 5 - y) into the second equation:
(5 - y) - 2y = -1
See what we did there? We replaced 'x' with its equivalent expression in terms of 'y'. This is where the magic of substitution happens! By replacing one variable with an expression involving the other, we've successfully transformed our two-variable system into a single-variable equation. This transformation is a cornerstone of many mathematical problem-solving techniques. It allows us to simplify complex relationships and focus on one unknown at a time. This step highlights the power of algebraic manipulation and sets the stage for us to solve for 'y'. The elegance of substitution lies in its ability to distill a multi-variable problem into a more manageable form. It's like breaking a complex puzzle into smaller, solvable pieces.
Step 3: Solve the new equation for 'y'.
Now we have a simple equation with just 'y':
5 - y - 2y = -1
Combine the 'y' terms:
5 - 3y = -1
Subtract 5 from both sides:
-3y = -6
Divide both sides by -3:
y = 2
Yay! We've found the value of 'y'! This is a significant milestone in our problem-solving journey. The isolation and solution of 'y' demonstrate the effectiveness of the substitution method. Each algebraic step we took was crucial in maintaining the equation's balance and leading us closer to the solution. It's a reminder that in mathematics, precision and attention to detail are paramount. Now that we have the value of 'y', we're just one step away from cracking the entire system. This success builds momentum and showcases the power of methodical problem-solving. The joy of solving for one variable fuels the anticipation of finding the other, completing our puzzle. So, let’s keep going – we're almost there!
Step 4: Substitute the value of 'y' back into either original equation to solve for 'x'.
Let's use the first equation, x + y = 5, and plug in y = 2:
x + 2 = 5
Subtract 2 from both sides:
x = 3
Solution:
So, we've found that x = 3 and y = 2. That means the solution to the system of equations is the ordered pair (3, 2). This is the point where the two lines intersect if we were to graph them. Congratulations! We've successfully navigated the substitution method and arrived at our solution. This achievement highlights the importance of methodical steps in problem-solving. By breaking down a complex problem into manageable parts, we can achieve clarity and accuracy. Remember, the solution (3, 2) is not just a pair of numbers; it's the harmonious meeting point of two equations. This visual interpretation adds depth to our understanding. We've not only solved the problem but also gained insight into the relationships between the variables. Now, let's explore another powerful method to solve systems of equations – the elimination method.
Method 2: Solving by Elimination
The elimination method, also known as the addition method, is another powerful technique for solving systems of equations. Instead of substituting, this method focuses on eliminating one of the variables by adding or subtracting the equations. The trick is to manipulate the equations so that the coefficients (the numbers in front of the variables) of one of the variables are opposites. When you add the equations together, that variable will cancel out, leaving you with a single equation in one variable. This is a really handy method when the equations are set up in a way that makes it easy to cancel out a variable. It’s like strategically combining forces to simplify the problem.
Let's use this method to solve our system:
- x + y = 5
- x - 2y = -1
Step 1: Multiply one or both equations by a constant so that the coefficients of one variable are opposites.
Notice that the 'x' terms in both equations already have the same coefficient (which is 1). To make them opposites, we can multiply the second equation by -1:
-1 * (x - 2y) = -1 * (-1)
This gives us:
-x + 2y = 1
Now our system looks like this:
- x + y = 5
- -x + 2y = 1
This step is a crucial setup for the elimination method. By strategically multiplying an equation, we align the coefficients of one variable to be opposites. This careful preparation is like setting up dominoes, where each piece is positioned to trigger the next. The goal here is to create an environment where adding the equations will cleanly eliminate a variable, making the subsequent steps much simpler. The skill in this step lies in choosing the right multiplier to achieve the desired cancellation. It requires a bit of foresight and a good understanding of algebraic manipulation. But once you get the hang of it, this technique becomes a powerful tool in your problem-solving arsenal.
Step 2: Add the equations together.
Now, we add the two equations together, aligning the 'x' terms, the 'y' terms, and the constants:
(x + y) + (-x + 2y) = 5 + 1
The 'x' terms cancel out (x - x = 0), which is exactly what we wanted!
This leaves us with:
3y = 6
The moment of truth in the elimination method! By carefully aligning and adding the equations, we've witnessed the elegant cancellation of one variable. This step is a testament to the power of strategic algebraic manipulation. It's like watching a perfectly executed magic trick, where a complex problem transforms into a simple equation. The cancellation of 'x' highlights the beauty of mathematical balance. We've successfully distilled our system into a single-variable equation, making it much easier to solve. This is a significant milestone in our journey, and it showcases the effectiveness of the elimination method. The sense of accomplishment here fuels our momentum as we move towards finding the value of 'y'. So, let’s keep the momentum going!
Step 3: Solve the resulting equation for 'y'.
Divide both sides of 3y = 6 by 3:
y = 2
We've found the value of 'y' again! It's always reassuring when different methods lead to the same answer. This step reinforces the reliability of mathematical principles. The simple division here belies the complex groundwork we laid in the previous steps. Solving for 'y' is like finding the missing piece of a puzzle. It brings us closer to completing the picture. This success boosts our confidence and reminds us that even seemingly complicated problems can be solved with methodical approaches. Now that we have the value of 'y', we're just one step away from fully unraveling the system. The anticipation is building as we prepare to find the value of 'x'. So, let's finish strong!
Step 4: Substitute the value of 'y' back into either original equation to solve for 'x'.
Let's use the first equation, x + y = 5, and plug in y = 2:
x + 2 = 5
Subtract 2 from both sides:
x = 3
Solution:
Again, we find that x = 3 and y = 2. The solution to the system is (3, 2). We've successfully solved the system using the elimination method, and our solution perfectly aligns with what we found using substitution. This consistency is a hallmark of sound mathematical problem-solving. Substituting the value of 'y' to find 'x' is the final stroke of the brush, completing our masterpiece. The solution (3, 2) represents the point of intersection of the two lines, a visual confirmation of our algebraic work. This step underscores the interconnectedness of mathematical concepts. We've not only solved the problem but also gained a deeper understanding of the relationships between the variables. The joy of reaching the final answer is amplified by the elegance and efficiency of the elimination method. We've added another valuable tool to our mathematical toolkit.
Conclusion
So, whether you prefer substitution or elimination, we've shown that both methods lead to the same solution for the system of equations x + y = 5 and x - 2y = -1, which is x = 3 and y = 2. The key takeaway here is that there's often more than one way to solve a math problem. The best method for you might depend on the specific problem or your personal preference. The beauty of mathematics lies in its flexibility and the multiple paths to discovery.
Understanding both substitution and elimination gives you a powerful arsenal for tackling systems of equations. Each method has its strengths and weaknesses, and knowing when to apply each one is a valuable skill. It's like having different gears on a bicycle – you can choose the one that's most efficient for the terrain. This mastery extends beyond the classroom, enhancing your problem-solving abilities in various aspects of life. We encourage you to practice these methods with different systems of equations. The more you practice, the more intuitive these techniques will become. Remember, math is not just about finding the answer; it's about developing a way of thinking. So, keep exploring, keep questioning, and keep solving!
I hope this explanation was helpful! Let me know if you have any other questions, guys, and keep practicing! You've got this! This journey through systems of equations has not only provided us with solutions but also sharpened our critical thinking skills. The ability to break down complex problems, choose appropriate methods, and execute them with precision is a valuable asset in any field. So, carry this mindset forward, and remember that every challenge is an opportunity to grow and learn. The world of mathematics is vast and exciting, filled with endless possibilities. Keep exploring, and you'll continue to uncover its beauty and power.