Solving 2x + 4y = 6 And X + Y = 2 A Step-by-Step Guide

Hey guys! Ever found yourself staring at a system of equations, feeling like you're trying to decipher ancient hieroglyphics? Well, you're not alone! Systems of equations can seem daunting at first, but trust me, they're totally solvable with the right approach. Today, we're going to break down a classic example: solving the system of equations 2x + 4y = 6 and x + y = 2. We'll explore different methods, step-by-step, so you can confidently tackle these problems. Whether you're a student prepping for an exam or just someone who loves a good math puzzle, this guide is for you. So, let's dive in and make those equations our friends!

Understanding Systems of Equations

Before we jump into the solution, let's take a moment to understand what a system of equations actually is. At its core, a system of equations is simply a set of two or more equations that share the same variables. In our case, we have two equations, and the variables are 'x' and 'y'. The goal is to find the values of these variables that satisfy all equations in the system simultaneously. Think of it like finding the sweet spot where all the equations agree. Graphically, this sweet spot represents the point where the lines corresponding to the equations intersect. This intersection point gives us the values of x and y that make both equations true. When you're dealing with these systems, remember that each equation represents a relationship between the variables, and we're trying to find the specific values that make all those relationships work together. This is a fundamental concept in algebra and has applications in various fields, from economics to engineering. So, mastering it is definitely a worthwhile endeavor. Now that we've got the basics down, let's move on to the fun part: solving our system!

Method 1: The Substitution Method

The substitution method is a powerful technique for solving systems of equations. The basic 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 one variable, which we can easily solve. Once we find the value of that variable, we can substitute it back into one of the original equations to find the value of the other variable. Let's see how this works with our example: 2x + 4y = 6 and x + y = 2. First, we need to choose an equation and isolate one of the variables. Looking at the second equation, x + y = 2, it seems easier to isolate 'x'. We can do this by subtracting 'y' from both sides, giving us x = 2 - y. Now, we have an expression for 'x' in terms of 'y'. The next step is to substitute this expression into the first equation, 2x + 4y = 6. Replacing 'x' with '(2 - y)', we get 2(2 - y) + 4y = 6. This is a single equation with only one variable, 'y'. We can now simplify and solve for 'y'. Distributing the 2, we get 4 - 2y + 4y = 6. Combining the 'y' terms, we have 4 + 2y = 6. Subtracting 4 from both sides gives us 2y = 2, and finally, dividing by 2, we find y = 1. Great! We've found the value of 'y'. Now, to find 'x', we substitute y = 1 back into the expression we found earlier, x = 2 - y. So, x = 2 - 1, which means x = 1. Therefore, the solution to the system of equations is x = 1 and y = 1. We can write this as an ordered pair (1, 1). This point represents the intersection of the two lines represented by the equations. And that's the substitution method in action! It's a versatile technique that can be applied to a wide range of systems of equations. Remember, the key is to choose the easiest variable to isolate and then carefully substitute and simplify.

Step-by-Step Solution using Substitution

Let's break down the substitution method for our system of equations (2x + 4y = 6 and x + y = 2) into clear, manageable steps. This way, you can see exactly how each move contributes to the final solution. 1. Isolate a Variable: Our first step is to pick one equation and solve for one of the variables. As we discussed earlier, the equation x + y = 2 is a good choice because it's relatively simple. Let's isolate 'x'. To do this, subtract 'y' from both sides of the equation: x + y - y = 2 - y. This simplifies to x = 2 - y. We now have 'x' expressed in terms of 'y'. 2. Substitute: The next crucial step is substitution. We'll take the expression we just found for 'x' (which is 2 - y) and substitute it into the other equation in the system, which is 2x + 4y = 6. This means replacing 'x' in the first equation with '(2 - y)'. So, we get: 2(2 - y) + 4y = 6. Notice that we now have a single equation with only one variable, 'y'. This is exactly what we wanted! 3. Solve for the Remaining Variable: Now, let's solve the equation we obtained in the previous step. First, distribute the 2: 2 * 2 - 2 * y + 4y = 6. This simplifies to 4 - 2y + 4y = 6. Next, combine the 'y' terms: 4 + 2y = 6. Subtract 4 from both sides: 2y = 2. Finally, divide both sides by 2: y = 1. We've successfully found the value of 'y'! 4. Substitute Back: We're not done yet! We still need to find the value of 'x'. To do this, we'll substitute the value we just found for 'y' (which is 1) back into one of the original equations or the expression we found for 'x'. The easiest option is usually the expression we found when isolating 'x': x = 2 - y. Substituting y = 1, we get: x = 2 - 1. This simplifies to x = 1. 5. Write the Solution: We've found both 'x' and 'y'! The solution to the system of equations is x = 1 and y = 1. We can write this as an ordered pair: (1, 1). This ordered pair represents the point where the lines corresponding to the two equations intersect on a graph. And there you have it! By following these steps carefully, you can confidently solve systems of equations using the substitution method. Remember to double-check your work and make sure your solution satisfies both original equations.

Method 2: The Elimination Method

Another fantastic method for tackling systems of equations is the elimination method. This approach focuses on eliminating one of the variables by manipulating the equations so that their coefficients are opposites. When you add the equations together, the variable with opposite coefficients cancels out, leaving you with a single equation with one variable. Let's see how this works with our trusty example: 2x + 4y = 6 and x + y = 2. Our goal is to make the coefficients of either 'x' or 'y' opposites in the two equations. Looking at the equations, it seems easier to eliminate 'x'. The coefficient of 'x' in the first equation is 2, and in the second equation, it's 1. To make them opposites, we can multiply the second equation by -2. This gives us: -2(x + y) = -2(2), which simplifies to -2x - 2y = -4. Now, we have the following system: 2x + 4y = 6 and -2x - 2y = -4. Notice that the coefficients of 'x' are now 2 and -2, which are opposites. The next step is to add the two equations together. Adding the left-hand sides, we get: (2x + 4y) + (-2x - 2y). Adding the right-hand sides, we get: 6 + (-4). Simplifying the left-hand side, the '2x' and '-2x' terms cancel out, leaving us with 2y. Simplifying the right-hand side, we get 2. So, our new equation is 2y = 2. Now, we can easily solve for 'y' by dividing both sides by 2, which gives us y = 1. Great! We've found the value of 'y'. To find 'x', we substitute y = 1 back into either of the original equations. Let's use the second equation, x + y = 2. Substituting y = 1, we get x + 1 = 2. Subtracting 1 from both sides, we find x = 1. Therefore, the solution to the system of equations is x = 1 and y = 1, or (1, 1). The elimination method is particularly useful when the coefficients of one of the variables are already multiples of each other or are easy to make multiples. By strategically manipulating the equations, we can eliminate a variable and solve for the other. It's another valuable tool in your equation-solving arsenal!

Step-by-Step Solution using Elimination

Let's walk through the elimination method step-by-step for our system of equations (2x + 4y = 6 and x + y = 2). This will solidify your understanding and help you apply this method with confidence. 1. Choose a Variable to Eliminate: First, we need to decide which variable we want to eliminate. In this case, 'x' seems like a good choice because its coefficients are relatively easy to manipulate. 2. Make Coefficients Opposites: To eliminate 'x', we need to make its coefficients opposites in the two equations. The coefficient of 'x' in the first equation is 2, and in the second equation, it's 1. To make them opposites, we can multiply the entire second equation by -2. This gives us: -2(x + y) = -2(2). Distribute the -2: -2x - 2y = -4. Now, we have the following system: 2x + 4y = 6 and -2x - 2y = -4. The coefficients of 'x' are now 2 and -2, which are opposites! 3. Add the Equations: The next step is to add the two equations together. This will eliminate the 'x' variable. Add the left-hand sides: (2x + 4y) + (-2x - 2y). Add the right-hand sides: 6 + (-4). 4. Simplify and Solve: Now, let's simplify the resulting equation. On the left-hand side, the '2x' and '-2x' terms cancel out, leaving us with: 2y. On the right-hand side, 6 + (-4) simplifies to 2. So, our equation is: 2y = 2. Divide both sides by 2 to solve for 'y': y = 1. We've successfully found the value of 'y'! 5. Substitute Back: To find 'x', we substitute the value of 'y' (which is 1) back into one of the original equations. Let's use the second equation, x + y = 2. Substitute y = 1: x + 1 = 2. Subtract 1 from both sides: x = 1. 6. Write the Solution: We've found both 'x' and 'y'! The solution to the system of equations is x = 1 and y = 1. We can write this as an ordered pair: (1, 1). This ordered pair represents the point where the lines corresponding to the two equations intersect. And that's the elimination method in action! By carefully manipulating the equations and adding them together, we eliminated a variable and solved for the other. This method is particularly powerful when the coefficients are easy to make opposites. Always remember to double-check your work to ensure your solution satisfies both original equations.

Verifying the Solution

Okay, guys, we've arrived at a solution using both the substitution and elimination methods, which is fantastic! But before we declare victory, there's one crucial step we absolutely must take: verifying the solution. This is like the final boss battle of solving systems of equations, and it ensures that our hard work has paid off and we haven't made any sneaky errors along the way. Verifying the solution is simple: we take the values we found for 'x' and 'y' and plug them back into the original equations. If both equations hold true, then our solution is correct. If even one equation fails, we know we need to go back and check our work for mistakes. Let's apply this to our solution of x = 1 and y = 1 for the system 2x + 4y = 6 and x + y = 2. First, let's plug the values into the first equation, 2x + 4y = 6. Substituting x = 1 and y = 1, we get: 2(1) + 4(1) = 6. Simplifying, we have: 2 + 4 = 6. And indeed, 6 = 6! So, the first equation holds true. Now, let's check the second equation, x + y = 2. Substituting x = 1 and y = 1, we get: 1 + 1 = 2. This simplifies to 2 = 2, which is also true! Since our solution satisfies both original equations, we can confidently say that x = 1 and y = 1 is the correct solution to the system. Verifying your solution is a non-negotiable step in solving systems of equations. It's a quick way to catch errors and ensure that your answer is accurate. Think of it as the ultimate safety net for your mathematical endeavors! So, always remember to verify – your future self will thank you.

Conclusion

Alright, guys, we've reached the end of our journey to solve the system of equations 2x + 4y = 6 and x + y = 2! We've explored two powerful methods – substitution and elimination – and walked through the solutions step-by-step. We also emphasized the importance of verifying our answers to ensure accuracy. Solving systems of equations might have seemed daunting at first, but hopefully, you now feel more confident in your ability to tackle these problems. Remember, the key is to break down the problem into manageable steps, choose the method that best suits the specific system, and always double-check your work. Mastering systems of equations is a valuable skill in mathematics and has applications in various fields. So, keep practicing, and you'll become a pro in no time! Whether you prefer the substitution method, the elimination method, or a combination of both, the important thing is to understand the underlying concepts and apply them systematically. And don't forget the final boss battle: verification! With practice and patience, you'll conquer any system of equations that comes your way. So go forth and solve, my friends! You've got this! And remember, math can actually be pretty fun when you break it down and understand the steps. Keep exploring, keep learning, and keep those equations in line!