Solving Systems of Linear Equations: A Comprehensive Guide
Systems of linear equations are a fundamental concept in algebra, often encountered in various fields such as engineering, economics, and physics. This article explores the method of solving a system of linear equations, specifically using the given example through the elimination and substitution methods.
Introduction to Linear Equations
A linear equation in three variables, such as (2x - y 3z 19), (5x - 2y 4z 13), and (3x 3y - z 2), can be solved by finding values of (x), (y), and (z) that satisfy all three equations simultaneously. This process is known as solving systems of linear equations.
Methods of Solving Systems of Linear Equations
1. Back Substitution Method
The back substitution method involves simplifying a system of equations by eliminating one variable at a time until only one equation and one variable remain. This method is particularly useful when dealing with smaller systems of equations.
2. Elimination Method
The elimination method involves combining two or more equations to eliminate one variable, typically by adding or subtracting them. This process is repeated until the system is reduced to a simpler form that can be solved more easily.
Example: Solving the Given System of Equations
Consider the following system of linear equations:
[2x - y 3z 19]
[5x - 2y 4z 13]
[3x 3y - z 2]
Step 1: Choosing the Pivot Equation
We start by selecting an equation as a pivot. For simplicity, let's choose the third equation:
[3x 3y - z 2]
Step 2: Eliminating a Variable
We can eliminate (z) by combining the first two equations with the pivot equation. To do this, we multiply the equations as follows:
[2x - y 3z 19 implies 6x - 3y 9z 57]
[3x 3y - z 2 implies 9x 9y - 3z 6]
Adding these two equations, we get:
[6x - 3y 9z 9x 9y - 3z 57 6]
[15x 6y 6z 63]
[5x 2y 21]
Similarly, we can eliminate (z) by combining the first and the second equations:
[5x - 2y 4z 13 implies 1 - 4y 8z 26]
[3x 3y - z 2 implies 9x 9y - 3z 6]
Multiplying the second equation by 2 to align with the first equation's coefficients, we get:
[1 - 4y 8z - 18x - 18y 6z 26 - 12]
[-8x - 22y 14z 14]
[-4x - 11y 7z 7]
We can eliminate (z) by subtracting this from the first equation adjusted for consistency:
[1 - 4y 8z - (9x 9y - 3z) 26 - 6]
[x - 13y 11z 20]
Step 3: Solving the Reduced System
Now we have a reduced system of two equations with two variables:
[5x 2y 21]
[-8x - 22y 14z 14]
We can solve the first equation for (x):
[5x 2y 21 implies x -frac{41}{13}]
Substituting (x) into the second equation, we get:
[68x - 40y 84 implies -41 - 40y 84 implies y frac{97}{13}]
Finally, substituting (x) and (y) into the pivot equation, we solve for (z):
[3 left(-frac{41}{13}right) 3 left(frac{97}{13}right) - z 2 implies -frac{123}{13} frac{291}{13} - z 2 implies z frac{142}{13}]
Step 4: Verify the Solution
Finally, we verify the solution by substituting (x -frac{41}{13}), (y frac{97}{13}), and (z frac{142}{13}) back into the original equations:
[2 left(-frac{41}{13}right) - frac{97}{13} 3 left(frac{142}{13}right) -frac{82}{13} - frac{97}{13} frac{426}{13} frac{247}{13} 19 text{ (True)}]
[5 left(-frac{41}{13}right) - 2 left(frac{97}{13}right) 4 left(frac{142}{13}right) -frac{205}{13} - frac{194}{13} frac{568}{13} frac{179}{13} 13 text{ (True)}]
[3 left(-frac{41}{13}right) 3 left(frac{97}{13}right) - frac{142}{13} -frac{123}{13} frac{291}{13} - frac{142}{13} frac{26}{13} 2 text{ (True)}]
Thus, the solution is verified to be correct.
Conclusion
Solving systems of linear equations requires careful application of algebraic methods. The provided example demonstrates the power and reliability of the elimination and back substitution methods. Understanding these methods can significantly enhance one's problem-solving skills in various mathematical and real-world applications.
Keywords:
- Linear Equations - System of Equations - Algebraic Methods