Math Problem Statement

how many independent and dependent variables

How many independent and dependent variables?

Solution

You provided a system of three linear equations:

1) x−y+z=21) \ x - y + z = 2 2) x+y−z=32) \ x + y - z = 3 3) −x+y+z=43) \ -x + y + z = 4

To determine how many independent and dependent variables exist, let's analyze the system.

  • The system has three variables: xx, yy, and zz.
  • The system also consists of three equations.

For the number of independent variables, we must see if the three equations are linearly independent. If they are, all variables will be independent, meaning there are no redundant (dependent) variables or equations.

Step 1: Write the system in matrix form.

The matrix form of the system is:

1 & -1 & 1 \\ 1 & 1 & -1 \\ -1 & 1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix}$$ ### Step 2: Determine the rank of the matrix. The rank of a matrix corresponds to the number of linearly independent rows, which tells us how many independent equations there are. If the rank equals the number of variables (3 in this case), all variables are independent. After performing row operations, it can be verified that the system has **rank 3**, meaning all three equations are independent. ### Conclusion: - **Independent variables**: All variables ($$x$$, $$y$$, and $$z$$) are independent. - **Dependent variables**: None. Would you like to go through the steps for solving the system of equations? Let me know if you want additional details or have any questions! Here are some questions for further exploration: 1. How do we check if a system of equations is consistent? 2. What are the different methods to solve a system of linear equations? 3. Can this system have more than one solution? 4. How do matrix operations relate to solving systems of equations? 5. What is the significance of the rank of a matrix? **Tip**: Always check the rank of the matrix when solving systems of linear equations to determine if the system has a unique solution.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations

Formulas

Ax = B (matrix form of a system of linear equations)
Rank of a matrix

Theorems

Rank-Nullity Theorem

Suitable Grade Level

Grades 11-12 and College

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