Solve X + Y = 3, 2X - 5Y = 36 By Elimination
Hey guys! Today, we're diving into a super useful math technique called the elimination method. This is a fantastic way to solve systems of equations, which might sound intimidating, but trust me, it's not! We're going to break down a specific example step-by-step: solving the system X + Y = 3 and 2X - 5Y = 36 using the elimination method. Get ready to become elimination method pros!
Understanding Systems of Equations
Before we jump into the solution, let's make sure we're all on the same page about what a system of equations actually is. Think of it like this: you have two or more equations, each with two or more variables (usually represented by letters like X and Y), and we're trying to find the values for those variables that make all the equations true at the same time. It's like solving a puzzle where each equation is a piece, and we need to fit them together perfectly.
In our case, we have two equations:
- X + Y = 3
- 2X - 5Y = 36
We need to find the values for X and Y that satisfy both of these equations simultaneously. There are a few methods to tackle this, and today we're focusing on the elimination method. Why? Because it's super efficient and can be a lifesaver when dealing with more complex systems. The core idea behind the elimination method is to manipulate the equations so that when we add them together, one of the variables cancels out, leaving us with a single equation with just one variable. This is a massive simplification, making the problem much easier to solve. Think of it as strategically subtracting or adding the equations to "eliminate" one variable, hence the name! This method is particularly handy when the coefficients (the numbers in front of the variables) are easy to work with or can be easily made to be opposites. So, let's see how this works in practice!
Step-by-Step Solution Using the Elimination Method
Alright, let's get our hands dirty and solve this system! The elimination method is all about making one of the variables disappear when we combine the equations. To do this, we need to make the coefficients of either X or Y opposites of each other. Looking at our system:
- X + Y = 3
- 2X - 5Y = 36
We can see that it would be relatively easy to make the X coefficients opposites. We can multiply the first equation by -2. This will give us -2X in the first equation, which is the opposite of the 2X in the second equation. Remember, whatever we do to one side of the equation, we have to do to the other to keep things balanced. So, let's multiply the entire first equation by -2:
-2 * (X + Y) = -2 * 3
This simplifies to:
-2X - 2Y = -6
Now, we have a modified system of equations:
- -2X - 2Y = -6
- 2X - 5Y = 36
Now comes the magic! We're going to add these two equations together. Notice what happens to the X terms: -2X + 2X = 0. The X variable is eliminated! This is exactly what we wanted. When we add the equations, we add the left-hand sides together and the right-hand sides together:
(-2X - 2Y) + (2X - 5Y) = -6 + 36
Simplifying this, we get:
-7Y = 30
Now we have a simple equation with just one variable, Y. To solve for Y, we simply divide both sides by -7:
Y = 30 / -7 Y = -30/7
Boom! We've found the value of Y. Now we need to find the value of X. We can do this by substituting the value of Y back into either of our original equations. Let's use the first equation, X + Y = 3, because it looks a little simpler.
Substituting Y = -30/7 into X + Y = 3, we get:
X + (-30/7) = 3
To solve for X, we add 30/7 to both sides:
X = 3 + 30/7
To add these, we need a common denominator. We can rewrite 3 as 21/7:
X = 21/7 + 30/7 X = 51/7
And there you have it! We've found the values of both X and Y. X = 51/7 and Y = -30/7. We've successfully solved the system of equations using the elimination method.
Checking Our Solution
It's always a good idea to double-check our work, especially in math. To do this, we'll substitute our values for X and Y back into both of the original equations to make sure they hold true. This is a crucial step to ensure we haven't made any mistakes along the way. First, let's check the equation X + Y = 3. We substitute X = 51/7 and Y = -30/7:
(51/7) + (-30/7) = 3
Simplifying the left side:
21/7 = 3
And indeed, 3 = 3. So, our solution works for the first equation. Great! Now, let's check the second equation, 2X - 5Y = 36. Again, we substitute X = 51/7 and Y = -30/7:
2 * (51/7) - 5 * (-30/7) = 36
Simplifying:
102/7 + 150/7 = 36
Combining the fractions:
252/7 = 36
And 252/7 does indeed equal 36. So, our solution also works for the second equation! This confirms that we've found the correct values for X and Y. Checking our solution not only gives us confidence in our answer but also helps us catch any potential errors. It's a small step that can save us a lot of headaches in the long run. So, remember to always check your solutions whenever you're solving systems of equations or any other math problem for that matter!
Why the Elimination Method Works
You might be wondering, why does this elimination method actually work? It seems a bit like mathematical magic, but there's a solid principle behind it. The key is that we're performing valid algebraic operations that don't change the fundamental solutions of the system. When we multiply an entire equation by a constant (like -2 in our example), we're essentially scaling both sides of the equation equally. This doesn't change the relationship between X and Y that satisfies the equation. Think of it like resizing a photograph – the proportions remain the same even if the size changes.
Similarly, when we add two equations together, we're essentially saying that if two things are equal (the left-hand sides of the equations) and two other things are equal (the right-hand sides of the equations), then adding the equal things together will still result in equal things. It's a logical extension of the basic principle of equality. The magic happens when we strategically manipulate the equations so that adding them eliminates one of the variables. This simplifies the problem into a single equation with one unknown, which we can easily solve. Once we find the value of one variable, we can substitute it back into any of the original equations to find the value of the other variable. The elimination method is a powerful tool because it allows us to systematically reduce a complex problem into simpler parts. It's a testament to the beauty and elegance of mathematics, where seemingly complex problems can be solved with a clever application of basic principles. So, the next time you use the elimination method, remember that you're not just following a set of rules, you're applying fundamental mathematical principles to unravel a puzzle!
When to Use the Elimination Method
The elimination method is a fantastic tool, but it's not always the best tool for every situation. So, when should you reach for this method in your mathematical toolbox? Generally, the elimination method shines when the coefficients of one of the variables in the system are either the same or easy to make the same (or opposites) by multiplication. This makes the process of eliminating a variable straightforward and efficient. For example, in our problem, the X coefficients were 1 and 2. It was easy to multiply the first equation by -2 to make the X coefficients opposites (-2 and 2). If the coefficients were something like 3 and 7, it would still be doable (you could multiply the first equation by 7 and the second by -3), but it would involve slightly larger numbers.
The elimination method is particularly advantageous when the equations are in standard form (Ax + By = C). This makes it easy to line up the variables and perform the addition or subtraction. However, if the equations are in a different form, like slope-intercept form (y = mx + b), the substitution method might be more convenient. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This is a great alternative when one of the equations is already solved for a variable or can be easily solved. Ultimately, the best method to use depends on the specific system of equations you're dealing with. It's like choosing the right tool for the job – a screwdriver is great for screws, but you wouldn't use it for a nail! With practice, you'll develop a sense for which method will be the most efficient for a given problem. And remember, knowing multiple methods gives you more flexibility and problem-solving power!
Practice Makes Perfect
The best way to master the elimination method, or any math technique for that matter, is to practice! Working through various examples will help you solidify your understanding of the steps involved and develop a feel for when this method is most effective. Don't be afraid to try different problems, even if they seem challenging at first. The more you practice, the more comfortable and confident you'll become. Start with simpler systems of equations where the coefficients are small and easy to manipulate. As you gain confidence, you can move on to more complex systems with larger numbers or fractions. Try creating your own systems of equations and solving them. This is a great way to test your understanding and identify any areas where you might need more practice. You can also find plenty of practice problems online or in textbooks. Look for problems that specifically ask you to use the elimination method. When you're working through problems, make sure to show your work clearly and step-by-step. This will help you track your progress and identify any mistakes you might be making. And don't be discouraged if you get stuck! Math can be challenging, but with perseverance and practice, you can overcome any obstacle. If you're struggling with a particular problem, try breaking it down into smaller steps, reviewing the concepts, or seeking help from a teacher, tutor, or classmate. Remember, everyone learns at their own pace, and the key is to keep practicing and never give up. So, grab a pencil, a piece of paper, and start practicing the elimination method today. You'll be solving systems of equations like a pro in no time!
Conclusion
So there you have it! We've successfully navigated the world of systems of equations using the elimination method. We've seen how to manipulate equations, eliminate variables, and solve for our unknowns. Remember, the elimination method is a powerful tool for solving systems of equations, especially when the coefficients are easy to work with. By following the steps outlined above and practicing regularly, you'll become a master of this technique. Keep practicing, and you'll be tackling even the most challenging systems of equations with confidence! Keep up the great work, guys!
