Solving Linear Equations: Find X1 - X2
Hey guys! Today, we're diving into the world of linear equations. We've got a system of three equations with three unknowns, and our mission is to find the value of x1 - x2. It might sound a bit daunting, but don't worry, we'll break it down step by step. Linear equations are fundamental in various fields, from engineering to economics, so mastering them is a valuable skill. This problem not only helps us practice solving these equations but also shows how to manipulate the solutions to find specific expressions. So, grab your thinking caps, and letβs get started!
Understanding the Problem
Before we jump into solving, let's make sure we understand what we're dealing with. We have a system of three linear equations:
Our goal is to find the values of x1, x2, and x3 that satisfy all three equations simultaneously. Once we have those values, we can easily calculate x1 - x2. There are several methods we can use to solve this system, including substitution, elimination, and matrix methods. We'll use the elimination method here because it's quite straightforward for this particular system. Remember, the key is to manipulate the equations in a way that allows us to eliminate one variable at a time, making the system simpler to solve.
Solving the System of Equations
Let's use the elimination method to solve this system. Our first goal is to eliminate one of the variables. Looking at the equations, x2 seems like a good candidate because the coefficients have opposite signs in the first two equations. This will make it easier to eliminate. The elimination method involves adding or subtracting multiples of equations to cancel out variables. The ultimate goal here is to simplify the system by reducing the number of variables in each equation until we can solve for them one by one.
Step 1: Eliminate x2 from Equations 1 and 2
Notice that the coefficients of x2 in the first two equations are 3 and -3, which are perfect for elimination. We can simply add the first and second equations together:
This simplifies to:
Let's call this new equation Equation 4.
Step 2: Eliminate x2 from Equations 2 and 3
Now, let's eliminate x2 from the second and third equations. To do this, we'll multiply the third equation by -3 to match the coefficient of x2 in the second equation (but with the opposite sign), then add the equations:
Multiply Equation 3 by -3:
Now add this to Equation 2:
This simplifies to:
Let's call this Equation 5.
Step 3: Solve for x1 and x3
Now we have two equations (Equation 4 and Equation 5) with two variables (x1 and x3):
To solve this system, we can subtract Equation 5 from Equation 4:
Now that we have x1, we can substitute it back into either Equation 4 or 5 to find x3. Let's use Equation 4:
Step 4: Solve for x2
Now that we have x1 and x3, we can substitute these values into any of the original equations to find x2. Let's use Equation 2:
So, we have found that x1 = 1, x2 = 3, and x3 = -2.
Calculating x1 - x2
Now that we have the values of x1 and x2, we can easily calculate x1 - x2:
Final Answer
So, the value of x1 - x2 is -2. Isn't it awesome when everything comes together like that? This problem showcases the power of systematic problem-solving in algebra. By breaking down a complex system of equations into smaller, manageable steps, we were able to find the solution relatively easily. Remember, practice is key! The more you work with these kinds of problems, the more comfortable and confident you'll become.
Why This Matters
Understanding how to solve systems of linear equations is crucial in many fields. Engineers use them to design structures and circuits, economists use them to model markets, and computer scientists use them in algorithms. The skills you've honed here β breaking down a problem, applying systematic methods, and paying attention to detail β are transferable to countless other situations.
Practice Makes Perfect
If you're looking to sharpen your skills, try tackling similar problems. You can change the coefficients or the number of equations to create new challenges. Also, explore different methods for solving linear systems, such as matrix methods or using software tools. The more tools you have in your toolbox, the better equipped you'll be to handle any problem that comes your way. Keep practicing, and you'll become a true algebra whiz!
Conclusion
We've successfully solved a system of linear equations and found that x1 - x2 = -2. Remember, the key to these problems is to break them down into smaller steps and stay organized. Algebra can be challenging, but with a bit of practice and the right approach, you can conquer any equation! Keep up the great work, guys, and happy solving!