Elimination Method: Solving Systems Of Equations
Have you ever encountered a situation where you have two equations with two unknowns, and you need to find the values that satisfy both equations simultaneously? That's where the elimination method comes in handy! It's a powerful algebraic technique used to solve systems of equations, and in this article, we'll dive deep into how it works, step by step. We will focus on applying this method to a specific problem, choosing the correct ordered pair that satisfies the given system of equations. Understanding the elimination method is crucial not only for academic success in mathematics but also for real-world applications where multiple variables and constraints are involved. For instance, in engineering, economics, and computer science, solving systems of equations is a fundamental skill. So, let's embark on this mathematical journey together and unlock the secrets of the elimination method!
The core idea behind the elimination method is to manipulate the equations in such a way that when you add or subtract them, one of the variables is eliminated. This leaves you with a single equation with a single variable, which is much easier to solve. Once you've found the value of one variable, you can substitute it back into one of the original equations to find the value of the other variable. This process effectively reduces a complex problem with two variables into a series of simpler problems involving just one variable. The beauty of the elimination method lies in its systematic approach, which makes it less prone to errors compared to other methods like substitution or graphing. Furthermore, it is particularly useful when dealing with larger systems of equations with three or more variables, where other methods might become cumbersome. So, whether you're a student tackling algebra problems or a professional working on complex models, the elimination method is a valuable tool to have in your mathematical arsenal.
Understanding the Elimination Method
The elimination method is a fantastic way to solve systems of equations, especially when you have two equations with two variables. The basic principle is to manipulate the equations so that when you add or subtract them, one of the variables cancels out. This leaves you with a single equation in one variable, which you can easily solve. Let's break down the steps with a clear and friendly approach.
Steps in the Elimination Method
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Align the Equations: First, make sure your equations are neatly aligned, with the x terms, y terms, and constants in their own columns. This makes it easier to see what you're working with. This step is crucial because it sets the stage for the subsequent operations. Misaligned equations can lead to confusion and errors, so take your time to ensure everything is in its proper place. Aligning the equations not only simplifies the visual aspect but also aids in the mental processing of the problem. By having the variables and constants lined up, you can quickly identify which terms can be added or subtracted to eliminate a variable. This alignment also helps in organizing your work, reducing the chances of overlooking a term or making a mistake in the calculations. In essence, proper alignment is the foundation for a successful application of the elimination method.
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Multiply (if necessary): Sometimes, the coefficients of the x or y terms aren't the same or opposites. If that's the case, you'll need to multiply one or both equations by a constant so that the coefficients of one variable are the same or additive inverses (opposites). This step is often the most critical in the elimination method because it sets up the actual elimination process. The goal is to make the coefficients of one of the variables either identical or opposite in sign. This way, when you add or subtract the equations, that variable will be eliminated. The choice of the multiplier depends on the specific equations you are dealing with. You may need to multiply one equation, or both, to achieve the desired coefficients. It's essential to choose the multipliers carefully to keep the arithmetic as simple as possible and minimize the chances of error. This step requires a bit of foresight and strategic thinking, but with practice, it becomes second nature.
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Add or Subtract: Now, add or subtract the equations. If you've set things up right, one of the variables should disappear, leaving you with an equation in just one variable. This is the heart of the elimination method. By carefully choosing the multipliers in the previous step, you've arranged for either the x or y terms to cancel out when the equations are added or subtracted. This simplifies the problem dramatically, reducing it from a system of two equations with two variables to a single equation with one variable. The decision to add or subtract depends on whether the coefficients of the variable you're trying to eliminate are the same or opposites. If they are the same, you subtract; if they are opposites, you add. This step requires attention to detail to ensure you are performing the correct operation and that you are accurately combining the terms. The result is a much simpler equation that you can solve easily.
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Solve: Solve the resulting equation for the remaining variable. Once you've eliminated one variable, the equation you're left with is straightforward to solve. It's typically a linear equation in one variable, which you can solve using basic algebraic techniques. This step is a direct application of the fundamental principles of algebra. You isolate the variable by performing the necessary operations, such as adding, subtracting, multiplying, or dividing both sides of the equation by a constant. The goal is to get the variable by itself on one side of the equation, revealing its value. This step provides the first piece of the puzzle in solving the system of equations. With the value of one variable in hand, you can move on to the next step, which involves finding the value of the other variable. The simplicity of this step underscores the power of the elimination method in transforming a complex problem into a series of manageable steps.
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Substitute: Take the value you just found and substitute it back into one of the original equations to solve for the other variable. This step is the final link in the chain of solving the system of equations. Now that you know the value of one variable, you can plug it back into either of the original equations to find the value of the other variable. The choice of which equation to use is arbitrary; either one will work. However, it's often wise to choose the equation that looks simpler or that will involve less arithmetic. Substitution is a powerful technique in algebra, and it's used extensively in various mathematical contexts. In this case, it allows you to leverage the value you've already found to unlock the value of the remaining variable. The result is a complete solution to the system of equations, with values for both variables. This final step brings closure to the problem-solving process.
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Check Your Solution: Always, always, always check your solution by plugging both values back into the original equations to make sure they work. This is the crucial verification step in the problem-solving process. After you've found the values of both variables, it's essential to check that these values satisfy both of the original equations. This ensures that you haven't made any errors in your calculations and that your solution is correct. To check your solution, substitute the values you've found for the variables into each of the original equations. If the equations hold true (i.e., the left-hand side equals the right-hand side), then your solution is correct. If not, you'll need to go back and review your steps to identify any mistakes. This step not only confirms the correctness of your solution but also reinforces your understanding of the problem-solving process. It's a good habit to develop, as it helps prevent careless errors and builds confidence in your mathematical abilities.
Solving the System of Equations
Now, let's apply these steps to the system of equations given:
egin{array}{l} x+y=6 \ x-y=4 \end{array}
Step-by-Step Solution
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Align the Equations: The equations are already neatly aligned:
egin{array}{l} x+y=6 \ x-y=4
\end{array}$
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Multiply (if necessary): Notice that the coefficients of y are already opposites (+1 and -1). So, we can skip this step!
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Add or Subtract: Add the two equations:
egin{array}{rcl} (x + y) + (x - y) &=& 6 + 4 \ 2x &=& 10
\end{array}$
The *y* terms cancel out, leaving us with an equation in *x*.
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Solve: Solve for x:
egin{array}{rcl} 2x &=& 10 \ x &=& 5
\end{array}$
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Substitute: Substitute x = 5 into the first equation:
egin{array}{rcl} 5 + y &=& 6 \ y &=& 1
\end{array}$
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Check Your Solution: Plug x = 5 and y = 1 back into both original equations:
- Equation 1: 5 + 1 = 6 (Correct!)
- Equation 2: 5 - 1 = 4 (Correct!)
Choosing the Correct Ordered Pair
We found that x = 5 and y = 1. This means the solution to the system of equations is the ordered pair (5, 1).
Looking at the options provided:
A. (5,1) B. (3,3) C. (2,4) D. (6,0)
The correct ordered pair is A. (5,1).
Conclusion
The elimination method is a powerful tool for solving systems of equations. By carefully aligning, multiplying, adding or subtracting, solving, substituting, and checking, you can confidently find the solutions to these types of problems. Remember, practice makes perfect! The more you use this method, the more comfortable and efficient you'll become. Don't be afraid to tackle complex problems; break them down into smaller, manageable steps, and you'll be solving systems of equations like a pro in no time! Understanding and mastering the elimination method opens doors to more advanced mathematical concepts and problem-solving techniques. It's a fundamental skill that will serve you well in various academic and professional contexts. So, keep practicing, keep exploring, and enjoy the journey of mathematical discovery!
For further reading and practice, you can visit Khan Academy's Systems of Equations for more examples and exercises. This is a valuable resource for anyone looking to deepen their understanding of this topic.
