Solving 2x-y=11 Using Elimination Method A Step-by-Step Guide
Understanding the Vilopan Vidhi (Elimination Method)
The Vilopan Vidhi, also known as the elimination method, is a fundamental algebraic technique used to solve systems of linear equations. These systems involve two or more equations with the same variables, and the goal is to find the values of those variables that satisfy all equations simultaneously. This method is particularly useful when dealing with two equations and two unknowns, as is the case in the example 2x - y = 11. The underlying principle of the elimination method is to manipulate the equations in such a way that, when added or subtracted, one of the variables is eliminated. This leaves us with a single equation in one variable, which can be easily solved. Once the value of this variable is found, it can be substituted back into one of the original equations to find the value of the other variable. The Vilopan Vidhi is a powerful tool in algebra and has wide applications in various fields, including mathematics, physics, engineering, and economics. It provides a systematic approach to solving linear systems and is especially useful when graphical methods or substitution methods become cumbersome or less efficient. The key to mastering the Vilopan Vidhi lies in understanding the underlying principles and practicing with a variety of examples. This will not only improve your algebraic skills but also enhance your problem-solving abilities in a broader context. The elimination method is a cornerstone of linear algebra, a branch of mathematics that deals with linear equations and their transformations. Linear algebra has applications in numerous fields, including computer graphics, data analysis, and optimization. By mastering the Vilopan Vidhi, you are building a strong foundation for further studies in mathematics and related disciplines.
Step-by-Step Solution for 2x - y = 11
To effectively demonstrate the Vilopan Vidhi, we need a system of two equations. Let's introduce a second equation to work with. For this example, we'll use the equation x + y = 4. This will allow us to illustrate the steps of the elimination method in a clear and concise manner. Now we have the following system of equations:
- 2x - y = 11
- x + y = 4
Now, let's walk through the steps of the Vilopan Vidhi.
Step 1: Align the Equations
The first critical step in using the Vilopan Vidhi is to ensure that the equations are properly aligned. This means writing the equations one below the other, with like terms (terms containing the same variable) in the same columns. This alignment is crucial for the subsequent steps, as it allows us to easily identify which variables can be eliminated by addition or subtraction. In our example, the equations are already aligned:
- 2x - y = 11
- x + y = 4
The x terms are aligned, the y terms are aligned, and the constant terms are aligned. This alignment sets the stage for the next step, which involves manipulating the equations to make the coefficients of one variable the same (or additive inverses) in both equations. Proper alignment not only makes the process clearer but also reduces the chances of making errors. It ensures that you are adding or subtracting the correct terms, which is essential for the elimination method to work effectively. If the equations were not initially aligned, you would need to rearrange them before proceeding further. This might involve moving terms from one side of the equation to the other, or simply rewriting the equations in a more organized manner.
Step 2: Identify the Variable to Eliminate
Identifying the variable you want to eliminate is a crucial strategic step in the Vilopan Vidhi. In this step, we carefully examine the coefficients of the variables in both equations to determine which variable would be easiest to eliminate. The goal is to look for variables whose coefficients are either the same or are additive inverses (i.e., have the same numerical value but opposite signs). This makes the elimination process simpler, as we can directly add or subtract the equations to eliminate the chosen variable.
In our example, we have the equations:
- 2x - y = 11
- x + y = 4
Looking at the coefficients, we can see that the coefficients of y are -1 and +1. These are additive inverses, which means that if we add the two equations together, the y terms will cancel each other out. This makes y the ideal candidate for elimination in this particular case. If the coefficients were not additive inverses, we would need to manipulate one or both equations to make them so. This often involves multiplying one or both equations by a constant. However, since we already have additive inverses for the y variable, we can proceed directly to the next step.
The decision of which variable to eliminate can also be influenced by other factors, such as the presence of fractions or decimals in the coefficients. If one variable has simpler coefficients, it might be easier to eliminate that variable, even if it requires a bit more manipulation of the equations. The key is to choose the variable that will lead to the most straightforward and efficient solution. With practice, you will develop a good intuition for identifying the best variable to eliminate in any given system of equations.
Step 3: Eliminate the Variable
Now that we've identified y as the variable to eliminate, the next step is to actually perform the elimination. Since the coefficients of y are -1 and +1, which are additive inverses, we can simply add the two equations together. This will cause the y terms to cancel out, leaving us with a single equation in x. Adding the equations:
2x - y = 11
+ (x + y = 4)
----------------
3x + 0 = 15
When we add the equations, the -y and +y terms cancel each other out, as intended. This leaves us with the equation 3x = 15. This is a much simpler equation than the original system, as it only involves one variable. Now, we can easily solve for x. This step is the heart of the Vilopan Vidhi, as it reduces the system of equations to a single equation in one variable. The ability to eliminate a variable is what makes this method so powerful and efficient. It allows us to break down a complex problem into a simpler one, which can be solved using basic algebraic techniques. The elimination step can also involve subtraction if the coefficients of the variable you want to eliminate are the same. In that case, you would subtract one equation from the other to eliminate the variable. The key is to perform the operation (addition or subtraction) that will cause the chosen variable to disappear from the equation. After eliminating the variable, you should always check your work to make sure that you have performed the addition or subtraction correctly. This will help you avoid errors and ensure that you arrive at the correct solution.
Step 4: Solve for the Remaining Variable
Following the elimination of y, we are left with the equation 3x = 15. This equation is a simple linear equation in one variable, which can be easily solved using basic algebraic techniques. To solve for x, we need to isolate x on one side of the equation. This can be done by dividing both sides of the equation by the coefficient of x, which is 3. Performing this division:
3x / 3 = 15 / 3
x = 5
Thus, we find that x = 5. This is the value of the variable x that satisfies both equations in the original system. Solving for the remaining variable is a crucial step in the Vilopan Vidhi, as it gives us one part of the solution to the system of equations. Once we have found the value of one variable, we can use it to find the value of the other variable. This is typically done by substituting the value of the solved variable back into one of the original equations.
Step 5: Substitute to Find the Other Variable
Now that we have found the value of x (x = 5), the next step is to substitute this value back into one of the original equations to find the value of y. We can choose either of the original equations; the result will be the same. However, it's often easier to choose the equation that looks simpler or has smaller coefficients. In this case, the second equation, x + y = 4, appears to be simpler. Substituting x = 5 into this equation:
5 + y = 4
To solve for y, we need to isolate y on one side of the equation. This can be done by subtracting 5 from both sides of the equation:
y = 4 - 5
y = -1
Therefore, the value of y is -1. This step is crucial because it completes the solution to the system of equations. We now have the values of both x and y, which together form the solution to the system.
Step 6: Verify the Solution
The final and arguably most important step in solving any system of equations is to verify the solution. This ensures that the values we have found for x and y actually satisfy both equations in the original system. To verify our solution, we substitute the values x = 5 and y = -1 into both original equations:
- Equation 1: 2x - y = 11
- Substituting: 2(5) - (-1) = 10 + 1 = 11 (Correct)
- Equation 2: x + y = 4
- Substituting: 5 + (-1) = 5 - 1 = 4 (Correct)
Since the values x = 5 and y = -1 satisfy both equations, we can confidently say that this is the correct solution to the system of equations. Verification is a critical step because it helps to catch any errors that may have occurred during the solution process. It ensures that the solution is accurate and reliable. By substituting the values back into the original equations, we are essentially checking that the solution is consistent with the given information. If the values do not satisfy both equations, it indicates that there is an error somewhere in the solution process, and we need to go back and review our steps. Verification is a fundamental aspect of problem-solving in mathematics and other disciplines. It is a way of ensuring that our answers are correct and that our reasoning is sound. By making verification a routine part of your problem-solving process, you can improve your accuracy and build confidence in your abilities.
Conclusion
In conclusion, the Vilopan Vidhi, or elimination method, is a powerful technique for solving systems of linear equations. By following the step-by-step guide outlined above, you can effectively solve equations like 2x - y = 11 (when paired with another equation). Remember to align the equations, identify the variable to eliminate, perform the elimination, solve for the remaining variable, substitute to find the other variable, and most importantly, verify your solution. This method is a fundamental tool in algebra and has wide applications in various fields. With practice, you can master the Vilopan Vidhi and confidently solve a wide range of linear systems. The key to success is to understand the underlying principles, follow the steps carefully, and always verify your answers. By doing so, you will not only improve your algebraic skills but also enhance your problem-solving abilities in general.
