System Of Equations: Consistent, Inconsistent, Or Dependent?

by Andrew McMorgan 61 views

Hey guys! Ever stumbled upon a system of equations and wondered what kind it is? Is it consistent, inconsistent, or dependent? Don't worry, it's not as intimidating as it sounds! In this article, we're diving deep into the fascinating world of linear equation systems. We'll take a look at how to identify each type using a real-world example. We'll focus on the system:

3x + y + 3z = 1
x + 2y - z = 2
2x - y + 4z = 4

So, let's grab our mathematical toolkits and get started!

Understanding Systems of Equations

Before we jump into solving, let's clarify what each system type means. This is crucial for correctly identifying the nature of our equations.

  • Consistent System: A consistent system of equations has at least one solution. This means there's at least one set of values for our variables (x, y, and z in this case) that satisfies all equations simultaneously. A consistent system can be further classified as either independent (having exactly one solution) or dependent (having infinitely many solutions).
  • Inconsistent System: An inconsistent system is a system that has no solutions. There's no combination of values for the variables that can satisfy all equations at the same time. These equations, in a sense, contradict each other.
  • Dependent System: A dependent system is a consistent system that has infinitely many solutions. This usually happens when equations are multiples of each other or can be derived from each other, indicating they represent the same underlying relationship.

Knowing these definitions is the first step. Now, let's explore the methods we can use to classify our system.

Methods to Identify System Types

There are several methods we can use to determine if a system of equations is consistent, inconsistent, or dependent. Let's explore a few popular ones:

  1. Gaussian Elimination (Row Echelon Form): Gaussian elimination is a systematic method for solving systems of linear equations. The main goal is to transform the system's augmented matrix into row echelon form (or reduced row echelon form). By analyzing the resulting matrix, we can easily identify the system type. If we get a row like [0 0 0 | non-zero], the system is inconsistent. If we have free variables (variables without a leading 1), the system is dependent. If we have a unique solution, it’s consistent and independent.
  2. Substitution: In the substitution method, we solve one equation for one variable and then substitute that expression into the other equations. This reduces the number of variables and equations until we can solve for the remaining variables. If we arrive at a contradiction (e.g., 0 = 1), the system is inconsistent. If we get an identity (e.g., 0 = 0), the system is dependent.
  3. Elimination (Addition/Subtraction): This method involves adding or subtracting multiples of equations to eliminate variables. The goal is to reduce the system to a simpler form where we can easily solve for the variables. Similar to substitution, contradictions indicate an inconsistent system, and identities point towards a dependent system.
  4. Determinants (for 2x2 and 3x3 systems): For smaller systems, we can use determinants. If the determinant of the coefficient matrix is non-zero, the system has a unique solution (consistent and independent). If the determinant is zero, the system is either inconsistent or dependent. We need further analysis to differentiate between those two.

For our example, Gaussian elimination will be the most efficient method, as it provides a clear path to classifying the system. So, let’s roll up our sleeves and get to work!

Applying Gaussian Elimination

To use Gaussian elimination, we first need to represent our system of equations as an augmented matrix. This matrix consists of the coefficients of the variables and the constants on the right-hand side of the equations.

For our system:

3x + y + 3z = 1
x + 2y - z = 2
2x - y + 4z = 4

The augmented matrix is:

[ 3  1  3 | 1 ]
[ 1  2 -1 | 2 ]
[ 2 -1  4 | 4 ]

Our goal is to transform this matrix into row echelon form. This means we want to get leading 1s (also called pivots) in each row and zeros below them. We'll use elementary row operations to achieve this. These operations include:

  • Swapping two rows.
  • Multiplying a row by a non-zero constant.
  • Adding a multiple of one row to another row.

Let's start by swapping Row 1 and Row 2 to get a 1 in the top-left position:

[ 1  2 -1 | 2 ]
[ 3  1  3 | 1 ]
[ 2 -1  4 | 4 ]

Next, we want to eliminate the 3 in the second row and the 2 in the third row. We can do this by performing the following operations:

  • R2 = R2 - 3 * R1
  • R3 = R3 - 2 * R1

This gives us:

[  1  2 -1 |  2 ]
[  0 -5  6 | -5 ]
[  0 -5  6 |  0 ]

Now, let's divide Row 2 by -5 to get a leading 1:

[ 1  2 -1 |  2 ]
[ 0  1 -6/5 |  1 ]
[ 0 -5  6 |  0 ]

Finally, we eliminate the -5 in the third row by performing:

  • R3 = R3 + 5 * R2

This results in:

[  1   2  -1 |  2 ]
[  0   1 -6/5 |  1 ]
[  0   0   0 |  5 ]

Identifying the System Type

Take a close look at the final row of our matrix: [ 0 0 0 | 5 ]. This row represents the equation:

0x + 0y + 0z = 5

Which simplifies to:

0 = 5

This is a clear contradiction! Zero cannot equal five. This contradiction tells us that the system of equations has no solution.

Therefore, the system is inconsistent. Boom! We've cracked the case.

Conclusion

So, the answer to our initial question is B. Inconsistent system. Identifying the type of system is a crucial step in solving linear equations. By using methods like Gaussian elimination, we can systematically analyze the equations and determine whether they have a solution, no solutions, or infinitely many solutions. You've got this, guys! Keep practicing, and you'll become system-solving pros in no time! Remember, math can be fun when you break it down step by step. Now you can confidently tackle similar problems and impress your friends with your newfound knowledge of linear systems! Keep exploring the world of mathematics, and who knows what other exciting discoveries you'll make?