Solving System Of Equations Expressing Solutions In Terms Of Parameter Z

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In mathematics, solving a system of equations is a fundamental skill with applications across various fields, including engineering, physics, economics, and computer science. This article provides a detailed walkthrough of solving a system of three linear equations with three variables. We'll explore the process of identifying whether a system has a unique solution, is inconsistent (no solution), or is dependent (infinitely many solutions). When a system is dependent, we'll learn how to express the solutions in terms of a parameter, typically 'z'. This guide aims to equip you with the knowledge and techniques to tackle similar problems confidently.

The System of Equations

Let's consider the following system of linear equations:

2x + y - z = 2
x - 3y + 2z = 1
7x - 7y + 4z = 7

Our goal is to find the values of x, y, and z that satisfy all three equations simultaneously. We'll use a combination of techniques, primarily Gaussian elimination, to achieve this.

Step 1: Gaussian Elimination - Transforming the System

Gaussian elimination is a systematic method for solving systems of linear equations. It involves transforming the system into an upper triangular form (also known as row-echelon form) using elementary row operations. These operations include:

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

1.1 Eliminate 'x' from the Second and Third Equations

Our first objective is to eliminate the 'x' term from the second and third equations. To do this, we'll use the first equation as our pivot.

  • Equation 1: 2x + y - z = 2
  • Equation 2: x - 3y + 2z = 1
  • Equation 3: 7x - 7y + 4z = 7

  1. Multiply Equation 2 by -2 and add it to Equation 1:

    -2 * (x - 3y + 2z) = -2 * 1

    -2x + 6y - 4z = -2

    Adding this to Equation 1:

    (2x + y - z) + (-2x + 6y - 4z) = 2 + (-2)

    7y - 5z = 0

  2. Multiply Equation 1 by -7/2 and add it to Equation 3:

    (-7/2) * (2x + y - z) = (-7/2) * 2

    -7x - (7/2)y + (7/2)z = -7

    Adding this to Equation 3:

    (7x - 7y + 4z) + (-7x - (7/2)y + (7/2)z) = 7 + (-7)

    -(21/2)y + (15/2)z = 0

Our system now looks like this:

2x + y - z = 2
7y - 5z = 0
-(21/2)y + (15/2)z = 0

1.2 Eliminate 'y' from the Third Equation

Next, we eliminate the 'y' term from the third equation using the second equation as the pivot.

  1. Multiply Equation 2 by 3/2 and add it to Equation 3:

    (3/2) * (7y - 5z) = (3/2) * 0

    (21/2)y - (15/2)z = 0

    Adding this to Equation 3:

    (-(21/2)y + (15/2)z) + ((21/2)y - (15/2)z) = 0 + 0

    0 = 0

This result, 0 = 0, indicates that the third equation is a linear combination of the first two equations. This means the system is dependent, and there are infinitely many solutions.

Our simplified system is now:

2x + y - z = 2
7y - 5z = 0
0 = 0

Step 2: Expressing Solutions in Terms of the Parameter 'z'

Since the system is dependent, we'll express the solutions in terms of a parameter, which we'll choose to be 'z'.

2.1 Solve for 'y' in Terms of 'z'

From the second equation, 7y - 5z = 0, we can solve for 'y':

7y = 5z

y = (5/7)z

2.2 Solve for 'x' in Terms of 'z'

Now, substitute the expression for 'y' into the first equation:

2x + y - z = 2

2x + (5/7)z - z = 2

2x - (2/7)z = 2

2x = 2 + (2/7)z

x = 1 + (1/7)z

Step 3: The General Solution

We have now expressed x and y in terms of the parameter z. We can write the general solution as:

  • x = 1 + (1/7)z
  • y = (5/7)z
  • z = z

This represents an infinite set of solutions, where each value of 'z' corresponds to a specific solution (x, y, z) that satisfies the original system of equations.

3.1 Representing the Solution Set

The solution set can be represented as an ordered triple:

(x, y, z) = (1 + (1/7)z, (5/7)z, z)

Where 'z' can be any real number. This means that there are infinitely many solutions to the system, each lying on a line in three-dimensional space.

Step 4: Verification (Optional but Recommended)

To ensure our solution is correct, we can substitute the expressions for x and y back into the original equations and verify that they hold true.

4.1 Verification with Equation 1: 2x + y - z = 2

2 * (1 + (1/7)z) + (5/7)z - z = 2

2 + (2/7)z + (5/7)z - z = 2

2 + (7/7)z - z = 2

2 + z - z = 2

2 = 2 (The equation holds true)

4.2 Verification with Equation 2: x - 3y + 2z = 1

(1 + (1/7)z) - 3 * ((5/7)z) + 2z = 1

1 + (1/7)z - (15/7)z + 2z = 1

1 - (14/7)z + 2z = 1

1 - 2z + 2z = 1

1 = 1 (The equation holds true)

4.3 Verification with Equation 3: 7x - 7y + 4z = 7

7 * (1 + (1/7)z) - 7 * ((5/7)z) + 4z = 7

7 + z - 5z + 4z = 7

7 + 0z = 7

7 = 7 (The equation holds true)

Since the expressions for x and y satisfy all three original equations, our solution is verified.

Conclusion

Solving systems of linear equations is a crucial skill in mathematics and its applications. In this article, we tackled a system of three equations with three variables using Gaussian elimination. We encountered a dependent system, characterized by infinitely many solutions, and learned how to express these solutions in terms of a parameter 'z'. This approach allows us to represent the entire solution set concisely. Remember, the key steps involve transforming the system into an upper triangular form, identifying the dependency, and expressing the variables in terms of the chosen parameter. By understanding and practicing these techniques, you'll be well-equipped to handle a wide range of system of equations problems.

Mastering the art of solving systems of equations is not just about finding numerical answers; it's about understanding the underlying relationships between variables and the geometric interpretation of these relationships. In the case of dependent systems, the solutions often represent lines or planes in multi-dimensional space, offering a deeper insight into the nature of the problem. Therefore, practice and conceptual understanding are essential for achieving proficiency in this area.

Further Exploration and Practice

To solidify your understanding, consider exploring additional examples of systems of equations, including those with unique solutions and inconsistent systems. Practice applying Gaussian elimination and other techniques, such as matrix methods (e.g., using the inverse of a matrix or Cramer's rule), to solve various types of systems. Furthermore, investigate the applications of systems of equations in real-world scenarios, such as circuit analysis, network flow problems, and economic modeling. This will not only enhance your problem-solving skills but also broaden your appreciation for the practical significance of this mathematical concept.

By engaging with these further explorations, you'll develop a more comprehensive and versatile approach to solving systems of equations, making you a more confident and capable problem-solver in mathematics and beyond. Remember, the journey of learning mathematics is a continuous process of exploration, practice, and application. Embrace the challenges, seek out new insights, and never stop questioning and learning.

This detailed guide has walked you through the process of solving a system of equations, particularly when the system is dependent. By understanding the steps and the underlying concepts, you can confidently approach similar problems and expand your mathematical toolkit.