Solve X=5y And X+y=60: A Step-by-Step Guide
Hey guys! Today, we're diving into the exciting world of solving systems of equations. Specifically, we're going to tackle a classic problem: solving the system where x equals 5y and x plus y equals 60. Don't worry if that sounds intimidating – we'll break it down step-by-step, making it super easy to understand. So, grab your pencils and 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. Think of it as a set of two or more equations that share the same variables. Our goal is to find the values of those variables that make all the equations true simultaneously. In our case, we have two equations (x=5y and x+y=60) and two variables (x and y). There are several methods to solve systems of equations, but today we're going to focus on the substitution method because it's particularly well-suited for this problem.
The substitution method, at its core, involves solving one equation for one variable and then substituting that expression into the other equation. This might sound a bit abstract, but it will become crystal clear as we work through our example. It’s like having a puzzle where you replace one piece with another that represents the same thing, simplifying the overall picture. This method is highly effective when one of the equations is already solved for one variable, or can be easily manipulated to do so. This is exactly what we have in our system, with the equation x=5y already expressing x in terms of y.
The beauty of systems of equations lies in their ability to model real-world situations. From calculating the break-even point in business to determining the trajectory of a rocket, these systems provide a powerful tool for problem-solving. Imagine, for instance, you're trying to figure out how many apples and bananas you can buy with a certain budget, given their individual prices and a constraint on the total number of fruits. This scenario can easily be represented as a system of equations, where the variables represent the quantities of apples and bananas. Mastering the techniques to solve these systems opens doors to a wide range of applications, making it a valuable skill in mathematics and beyond.
Step 1: Identify the Equations
Okay, first things first, let's clearly identify our equations. We have:
- x = 5y
- x + y = 60
This is our starting point. We need to find the values of 'x' and 'y' that satisfy both of these equations. Notice how the first equation, x = 5y, tells us that x is directly related to y. Specifically, x is always five times the value of y. This relationship is crucial for our next step. The second equation, x + y = 60, introduces another constraint: the sum of x and y must equal 60. These two pieces of information, combined, will lead us to a unique solution for x and y.
Think of each equation as a piece of a puzzle. The first equation tells us something about the relationship between x and y, while the second equation gives us a different piece of information about their combined value. To solve the system, we need to put these pieces together in a way that reveals the solution. That's where the substitution method comes in handy. By using the first equation to express x in terms of y, we can substitute that expression into the second equation. This will transform the second equation into one with only one variable (y), which we can then easily solve.
Recognizing the structure of the equations is a critical first step in solving any system. Are the equations linear? Are they in slope-intercept form? Is one variable already isolated? Answering these questions will help you choose the most efficient solution method. In our case, the equations are linear (meaning they represent straight lines when graphed), and the first equation is already conveniently solved for x. This makes the substitution method an excellent choice. By understanding the properties of the equations, we can strategize our approach and avoid unnecessary complications.
Step 2: Substitute
This is where the magic happens! Since we know that x = 5y, we can substitute 5y in place of x in the second equation. This gives us:
(5y) + y = 60
See what we did there? We replaced the 'x' in the second equation with its equivalent value, '5y', from the first equation. This is the core idea of the substitution method. We've effectively eliminated one variable (x) from the second equation, leaving us with an equation that only involves 'y'. This is a significant step because we can now solve this single equation for 'y'.
The act of substitution is like performing a clever trade. We're trading 'x' for '5y', knowing that they represent the same value according to our first equation. This trade transforms the problem from a system of two equations to a single equation, which is much easier to handle. This is a common strategy in mathematics: simplify the problem by reducing the number of unknowns. By carefully substituting, we maintain the integrity of the equation while making it more manageable.
Substitution is not just a mathematical trick; it's a powerful problem-solving technique that can be applied in various contexts. Think about replacing a worn-out part in a machine with a new one, or substituting an ingredient in a recipe with a similar alternative. The underlying principle is the same: replace something with its equivalent to achieve a desired outcome. In our case, the desired outcome is to simplify the equation and make it solvable. The substitution step is a crucial bridge that connects the two equations and paves the way for finding the values of x and y.
Step 3: Solve for y
Now we have a simpler equation: 5y + y = 60. Let's solve for 'y'.
Combine the 'y' terms: 6y = 60
Divide both sides by 6: y = 10
Woohoo! We found the value of y. It's 10. This means that one piece of our puzzle is now in place. We know that 'y' must be 10 for both equations to be true simultaneously. This is a major breakthrough because now we can use this value to find the value of 'x'.
The process of solving for 'y' involved simplifying the equation by combining like terms and then isolating 'y' by performing the same operation on both sides of the equation. This is a fundamental technique in algebra, based on the principle that if you do the same thing to both sides of an equation, you maintain the equality. It's like balancing a scale: if you add or remove weight from one side, you need to do the same on the other side to keep it balanced.
Finding the value of 'y' is a significant milestone in solving the system of equations. It's like finding a key that unlocks the rest of the solution. Now that we know 'y', we can use this information to backtrack and find 'x'. This is where the power of systems of equations becomes truly apparent. The interconnectedness of the equations allows us to use the value of one variable to determine the value of the other. This is a testament to the elegance and efficiency of mathematical methods in solving complex problems.
Step 4: Solve for x
Great! We know y = 10. Now we can plug this value back into either of our original equations to solve for x. The easiest one to use is x = 5y. So:
x = 5 * 10 x = 50
Boom! We've found x. It's 50. We've now successfully found the values of both x and y that satisfy our system of equations. This is the culmination of our efforts, and it's a rewarding feeling to have cracked the code. Remember, the solution to a system of equations is a set of values that make all the equations true simultaneously.
Substituting the value of 'y' back into one of the original equations is a crucial step in completing the solution. It's like connecting the dots: we found 'y', and now we're using it to find 'x'. This process highlights the interconnectedness of the variables in a system of equations. Each variable depends on the others, and finding the value of one variable often leads to finding the values of the rest.
Choosing the easiest equation to substitute into is a smart strategy. In our case, x = 5y is simpler than x + y = 60, so it's the more efficient choice. This demonstrates the importance of strategic thinking in problem-solving. By carefully selecting the right approach, we can minimize the amount of work required and reduce the chances of making errors. The ability to choose the most efficient path is a valuable skill not only in mathematics but also in many other areas of life.
Step 5: Check Your Solution
This is super important, guys! Always check your solution to make sure it works. Plug x = 50 and y = 10 into both original equations:
- x = 5y -> 50 = 5 * 10 -> 50 = 50 (Yep!)
- x + y = 60 -> 50 + 10 = 60 -> 60 = 60 (Double yep!)
Our solution checks out! We know we've found the correct values for x and y. This step is crucial because it ensures that our solution is valid and that we haven't made any mistakes along the way. It's like proofreading an essay or testing a software program: it's a final check to catch any errors before they cause problems.
The process of checking the solution involves substituting the values of the variables back into the original equations and verifying that the equations hold true. This is a powerful way to ensure the accuracy of our work. It's not enough to simply find a solution; we must also confirm that it is indeed the correct solution. This step demonstrates the importance of rigor and attention to detail in mathematics and other disciplines.
Checking the solution also provides a deeper understanding of the system of equations. It reinforces the idea that the solution must satisfy all the equations simultaneously. If the solution doesn't work in even one of the equations, then it's not a valid solution. This reinforces the interconnectedness of the equations and the importance of finding values that work for the entire system. By taking the time to check our solution, we gain confidence in our answer and a better understanding of the underlying mathematical concepts.
Conclusion
And there you have it! We've successfully solved the system of equations x = 5y and x + y = 60. We found that x = 50 and y = 10. Remember, the key is to break down the problem into manageable steps, use the substitution method effectively, and always check your work. Solving systems of equations might seem daunting at first, but with practice, you'll become a pro in no time! Keep practicing, and you'll be tackling even more challenging problems in the future.
Solving systems of equations is a fundamental skill in mathematics with applications in various fields, from engineering to economics. The techniques we've discussed today, such as the substitution method, are powerful tools for solving these problems. By mastering these techniques, you'll be well-equipped to tackle a wide range of mathematical challenges. Remember, the key to success is practice. The more you practice, the more comfortable and confident you'll become in solving systems of equations.
So, what's next? Challenge yourself with more complex systems of equations. Explore other methods for solving systems, such as elimination and graphing. The world of mathematics is vast and fascinating, and solving systems of equations is just one piece of the puzzle. Keep exploring, keep learning, and keep challenging yourself. You'll be amazed at what you can achieve. Remember, every problem you solve is a step forward on your mathematical journey.
