Solving Linear Systems Finding Solution Sets, Inconsistent Systems, And Dependent Equations

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In mathematics, solving linear systems is a fundamental skill with applications across various fields, from engineering to economics. A linear system is a set of linear equations with the same variables. The solution set of a linear system is the set of all values for the variables that satisfy all equations simultaneously. In this article, we will delve into the methods for finding the solution set of a linear system, focusing on identifying inconsistent systems and dependent equations. We will use the following linear system as a case study:

\left\{\begin{aligned}
x+y & =0 \\
y+3 z & =-4 \\
y+z & =5-x
\end{aligned}\right.

This system consists of three equations with three variables: x, y, and z. Our goal is to find the values of these variables that satisfy all three equations.

Understanding Linear Systems

Before diving into the solution process, let's clarify some key concepts related to linear systems.

  • Linear Equation: A linear equation is an equation in which the highest power of any variable is 1. For example, x + y = 0 and y + 3z = -4 are linear equations.
  • System of Equations: A system of equations is a set of two or more equations with the same variables.
  • Solution Set: The solution set of a system of equations is the set of all values for the variables that satisfy all equations simultaneously. A solution is typically represented as an ordered tuple, such as (x, y, z).
  • Inconsistent System: An inconsistent system is a system of equations that has no solution. This means there are no values for the variables that can satisfy all equations simultaneously.
  • Dependent Equations: Dependent equations are equations that provide redundant information. In a system of dependent equations, one equation can be derived from the others. This means that one or more equations do not contribute unique information to the system.

Methods for Solving Linear Systems

There are several methods for solving linear systems, including:

  • Substitution: This method involves solving one equation for one variable and substituting that expression into the other equations.
  • Elimination: This method involves adding or subtracting multiples of the equations to eliminate one or more variables.
  • Gaussian Elimination: This is a systematic method for solving linear systems using row operations on the augmented matrix of the system.
  • Matrix Methods: Linear systems can also be solved using matrix operations, such as finding the inverse of the coefficient matrix or using Cramer's Rule.

In this article, we will primarily use the elimination and substitution methods to solve the given linear system.

Step-by-Step Solution

Let's solve the given linear system step-by-step:

\left\{\begin{aligned}
x+y & =0  &&(1)\\
y+3 z & =-4 &&(2)\\
y+z & =5-x &&(3)
\end{aligned}\right.

Step 1: Simplify the Equations

First, let's simplify equation (3) by adding x to both sides:

x + y + z = 5   (4)

Now we have the following system:

\left\{\begin{aligned}
x+y & =0  &&(1)\\
y+3 z & =-4 &&(2)\\
x + y + z & = 5   &&(4)
\end{aligned}\right.

Step 2: Use Elimination to Reduce the System

We can use equation (1) to eliminate x from equation (4). Subtract equation (1) from equation (4):

(x + y + z) - (x + y) = 5 - 0
z = 5   (5)

Now we know that z = 5. Let's substitute this value into equation (2):

y + 3(5) = -4
y + 15 = -4
y = -19   (6)

Step 3: Substitute to Find the Remaining Variable

Now that we have y = -19, we can substitute this value into equation (1) to find x:

x + (-19) = 0
x = 19   (7)

Step 4: Verify the Solution

We have found the solution x = 19, y = -19, and z = 5. Let's verify that this solution satisfies all three original equations:

  • Equation (1): 19 + (-19) = 0 (True)
  • Equation (2): -19 + 3(5) = -19 + 15 = -4 (True)
  • Equation (3): -19 + 5 = 5 - 19 => -14 = -14 (True)

The solution (x, y, z) = (19, -19, 5) satisfies all three equations.

Identifying Inconsistent Systems and Dependent Equations

In this case, we found a unique solution for the system, which means the system is consistent and the equations are independent. Let's discuss how to identify inconsistent systems and dependent equations.

Inconsistent Systems

An inconsistent system has no solution. This typically occurs when the equations in the system contradict each other. For example, consider the following system:

\left\{\begin{aligned}
x + y & = 1 \\
x + y & = 2
\end{aligned}\right.

This system is inconsistent because there are no values of x and y that can simultaneously satisfy both equations. If we subtract the first equation from the second, we get 0 = 1, which is a contradiction. Inconsistent systems will often lead to such contradictions when you attempt to solve them using substitution or elimination.

Dependent Equations

Dependent equations provide redundant information. This means that at least one equation can be derived from the others. For example, consider the following system:

\left\{\begin{aligned}
x + y & = 1 \\
2x + 2y & = 2
\end{aligned}\right.

The second equation is simply a multiple of the first equation (multiply the first equation by 2). This means that the second equation does not provide any new information. When solving systems with dependent equations, you will often find that you have fewer independent equations than variables, which leads to infinitely many solutions or a solution set that can be expressed in terms of parameters.

For instance, in the above example, we can express y in terms of x as y = 1 - x. The solution set is then all pairs (x, 1 - x) for any real number x. This represents a line in the xy-plane.

General Steps for Solving Linear Systems

To summarize, here are the general steps for solving a linear system:

  1. Simplify the equations: Remove any parentheses or fractions and combine like terms.
  2. Choose a method: Select either substitution or elimination (or Gaussian elimination for larger systems).
  3. Apply the method:
    • Substitution: Solve one equation for one variable and substitute that expression into the other equations.
    • Elimination: Add or subtract multiples of the equations to eliminate one or more variables.
  4. Solve for the remaining variables: Continue the process until you have solved for all variables.
  5. Verify the solution: Substitute the values you found back into the original equations to ensure they are satisfied.
  6. Identify the type of system:
    • Consistent and Independent: A unique solution exists.
    • Inconsistent: No solution exists.
    • Consistent and Dependent: Infinitely many solutions exist.

Conclusion

In conclusion, finding the solution set of a linear system involves systematically solving the equations to determine the values of the variables that satisfy all equations simultaneously. The given system:

\left\{\begin{aligned}
x+y & =0 \\
y+3 z & =-4 \\
y+z & =5-x
\end{aligned}\right.

has a unique solution (x, y, z) = (19, -19, 5). This system is consistent and the equations are independent. By understanding the methods for solving linear systems and how to identify inconsistent systems and dependent equations, you can tackle a wide range of mathematical problems and real-world applications. The key is to practice and apply these techniques methodically to ensure accurate and efficient solutions.

This comprehensive guide has provided a detailed explanation of how to solve linear systems, identify different types of systems, and understand the underlying concepts. By following these steps and practicing regularly, you can master the art of solving linear systems.

Keywords: Linear systems, solution set, inconsistent system, dependent equations, elimination method, substitution method, Gaussian elimination, matrix methods, consistent system, independent equations, linear equations, system of equations.