Solve Equations By Elimination: A Clear Guide
Hey guys! Today, we're diving deep into the world of linear equations and tackling a common question: how to solve a system of equations using the elimination method. We'll break down a specific problem step-by-step, making sure you grasp the core concepts and can confidently apply them to any similar challenge. So, let's get started!
Understanding the Elimination Method
The elimination method is a powerful technique for solving systems of linear equations. The main idea? To eliminate one of the variables by strategically adding or subtracting the equations. This leaves us with a single equation in one variable, which is super easy to solve. Once we've found the value of that variable, we can plug it back into either of the original equations to find the value of the other variable.
The elimination method is particularly useful when the coefficients of one of the variables are the same or easily made the same (or opposites) in the two equations. This allows for straightforward addition or subtraction to eliminate that variable. The beauty of the elimination method lies in its systematic approach, transforming a potentially complex problem into a series of manageable steps. It's a fundamental tool in algebra and essential for tackling more advanced mathematical concepts. So, buckle up, and let's learn how to wield this powerful method!
The Problem: Setting the Stage
Let's consider the following system of linear equations:
4x + 8y = 20
-4x + 2y = -30
Our mission? To find the values of x and y that satisfy both equations simultaneously. We'll walk through the elimination method step-by-step, explaining each move along the way. It’s crucial to understand not just the how but also the why behind each step. This deeper understanding is what transforms rote memorization into genuine problem-solving ability. Think of it like building a house; you need a strong foundation (understanding the principles) before you can start adding the walls and roof (solving the problem).
Before we jump into the solution, let's take a moment to observe the system. Notice anything special about the coefficients of x? They're 4 and -4 – opposites! This is a huge hint that the elimination method is going to be our best friend here. Recognizing these patterns is key to choosing the most efficient solution strategy. With a little practice, you'll start seeing these opportunities in all sorts of problems!
Step 1: Eliminate 'x'
The brilliance of this system lies in the coefficients of x. We have 4x in the first equation and -4x in the second. What happens when we add these terms together? They cancel each other out! This is the core of the elimination method in action.
So, let's add the two equations together, term by term:
(4x + 8y) + (-4x + 2y) = 20 + (-30)
Simplifying this, we get:
10y = -10
Notice how the x terms have vanished, leaving us with a single equation in y. This is exactly what we wanted! By adding the equations, we've effectively eliminated one variable and simplified the problem significantly. This step is the heart of the elimination method, and recognizing opportunities for this kind of cancellation is key to mastering the technique. It's like a magic trick, but with math!
Step 2: Solve for 'y'
Now we have a simple equation: 10y = -10. Solving for y is a breeze! We just need to divide both sides of the equation by 10:
y = -10 / 10
y = -1
Ta-da! We've found the value of y. This is a significant milestone. We've successfully navigated the first part of the puzzle. It's important to take a moment to appreciate this victory before moving on. Each step we take builds upon the previous one, and solving for y opens the door to finding x. Remember, math is a journey, not just a destination. Savor the small wins along the way!
Step 3: Substitute 'y' to Find 'x'
Now that we know y = -1, we can substitute this value into either of the original equations to solve for x. Let's use the first equation, 4x + 8y = 20:
4x + 8(-1) = 20
4x - 8 = 20
Now, we have a simple equation in x. Let's add 8 to both sides:
4x = 28
And finally, divide both sides by 4:
x = 7
We've done it! We've found the value of x. Substitution is a powerful technique that allows us to connect the values of different variables within a system of equations. It's like having a key that unlocks the remaining unknowns once we've solved for one variable. Practice with substitution will make it second nature, allowing you to efficiently solve a wide range of problems.
Step 4: The Solution
We've found that x = 7 and y = -1. So, the solution to the system of equations is the ordered pair (7, -1). This means that the point (7, -1) is the intersection of the two lines represented by the equations. Geometrically, this is where the two lines cross on a graph. Algebraically, it's the unique pair of values that satisfies both equations simultaneously.
It's always a good idea to check your solution by plugging the values of x and y back into both original equations to make sure they hold true. This helps catch any potential errors and gives you confidence in your answer. Consider it the final polish on your mathematical masterpiece! In this case:
- For
4x + 8y = 20:4(7) + 8(-1) = 28 - 8 = 20(Correct!) - For
-4x + 2y = -30:-4(7) + 2(-1) = -28 - 2 = -30(Correct!)
Our solution checks out! We've successfully solved the system of equations using the elimination method. Give yourself a pat on the back!
The Correct Statement
Based on our step-by-step solution, the correct statement explaining how Dana can solve the system of linear equations for x using the elimination method is:
- Add the two equations, solve for y, and then substitute the value of y into either of the original equations to solve for x.
This statement accurately captures the essence of the elimination method: adding the equations to eliminate a variable, solving for the remaining variable, and then substituting to find the other variable. It's a concise and clear description of the process we've just walked through. Understanding this core principle is key to applying the elimination method effectively in various situations.
Key Takeaways and Tips
- Look for opportunities to eliminate a variable: When the coefficients of one variable are the same or opposites, elimination is a great choice.
- Be careful with signs: Pay close attention to the signs when adding or subtracting equations.
- Substitute carefully: Make sure you substitute the correct value into the correct equation.
- Check your solution: Always plug your solution back into the original equations to verify your answer.
- Practice makes perfect: The more you practice, the more comfortable you'll become with the elimination method and the faster you'll be able to solve systems of equations.
Practice Problems
To solidify your understanding, try solving these systems of equations using the elimination method:
2x + y = 7x - y = 23x - 2y = 8x + 2y = 05x + 3y = 162x - 3y = 5
Working through these problems will help you develop your skills and build confidence in your ability to tackle linear equations. Remember, math is like a muscle; the more you exercise it, the stronger it gets! So, grab a pencil and paper, and get ready to conquer some equations!
Conclusion
The elimination method is a fantastic tool for solving systems of linear equations. By understanding the steps and practicing regularly, you'll become a pro at solving these problems. Remember, math is a journey of discovery, and each problem you solve is a step forward. Keep exploring, keep learning, and keep those mathematical muscles strong! You've got this!
