A Complete Guide to One Solution Method for Solving SPL
Solving systems of linear equations (SPL) is a fundamental concept in mathematics with applications across various fields like engineering, economics, and computer science. While multiple methods exist, one particularly efficient and elegant approach is the One Solution Method. This method focuses on systematically manipulating the equations to isolate one variable and then substitute its value back into the other equations to solve for the remaining variables. Let's delve into this powerful technique with clear examples.
Understanding the One Solution Method
The core principle behind this method is to reduce the complexity of the system by strategically eliminating variables. We achieve this by expressing one variable in terms of others from one equation and then substituting this expression into the remaining equations. This process continues until we have a single equation with only one variable, which is easily solved. The solution for this variable is then back-substituted into the previous equations to find the values of the other variables.
This method is particularly effective for systems with two or three equations and relatively simple coefficients. However, its application can become more complex with larger systems. For larger systems, matrix methods like Gaussian elimination or Cramer's rule might be more efficient.
Step-by-Step Procedure
Let's break down the One Solution Method into a clear, step-by-step procedure:
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Choose an Equation and Variable: Select one of the equations in the system and choose a variable that seems easiest to isolate (usually a variable with a coefficient of 1 or -1).
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Isolate the Chosen Variable: Manipulate the selected equation algebraically to express the chosen variable in terms of the other variables.
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Substitute: Substitute the expression obtained in step 2 into the remaining equations of the system. This will reduce the number of variables in those equations.
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Solve the Reduced System: Solve the resulting reduced system. This may involve repeating steps 1-3 until you reach a single equation with a single variable.
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Back-Substitute: Substitute the solution obtained in step 4 back into the previously solved equations to find the values of the remaining variables.
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Verify: Always check your solution by substituting the values of the variables into the original equations to ensure they satisfy all equations simultaneously.
Example: Solving a System of Two Linear Equations
Let's consider the following system of two linear equations:
- 2x + y = 5
- x - y = 1
Steps:
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Isolate: Let's isolate 'x' from the second equation: x = 1 + y
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Substitute: Substitute this expression for 'x' into the first equation: 2(1 + y) + y = 5
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Solve: Simplify and solve for 'y': 2 + 2y + y = 5 => 3y = 3 => y = 1
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Back-Substitute: Substitute y = 1 back into the expression for x: x = 1 + 1 = 2
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Solution: The solution to the system is x = 2 and y = 1.
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Verify: Substitute x = 2 and y = 1 into the original equations:
- 2(2) + 1 = 5 (True)
- 2 - 1 = 1 (True)
Therefore, our solution is correct.
Example: Solving a System of Three Linear Equations
The One Solution Method can also be extended to solve systems with three or more variables. The process is the same: isolate one variable, substitute, and repeat until you find all the variables' values. This can become more complex, requiring careful organization and attention to detail.
Conclusion
The One Solution Method provides a systematic and understandable approach to solving systems of linear equations, particularly those with a smaller number of variables. By following the steps carefully and verifying your solutions, you can confidently apply this method to various mathematical problems. Remember, for larger systems, consider using more advanced techniques. Understanding this method builds a strong foundation for more complex linear algebra concepts.
