Math Problem Statement

The system of equations 2x + 4y + 5z + w = 0 x - y + 7z + 2w = 0 9x - 7y + 8z + 16w = 0 has - No solutions Unique solution Infinitely many solutions Only trivial solution

Solution

To determine the type of solution for the given system of equations:

The system is:

  1. 2x+4y+5z+w=02x + 4y + 5z + w = 0
  2. xy+7z+2w=0x - y + 7z + 2w = 0
  3. 9x7y+8z+16w=09x - 7y + 8z + 16w = 0

This is a homogeneous system of linear equations. Homogeneous systems always have at least one solution, known as the trivial solution (x=0,y=0,z=0,w=0x = 0, y = 0, z = 0, w = 0).

Steps:

  1. Write the system in matrix form:

    2 & 4 & 5 & 1 \\ 1 & -1 & 7 & 2 \\ 9 & -7 & 8 & 16 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$
  2. Check the rank of the coefficient matrix. The rank determines the number of independent equations, which decides the type of solution:

    • If the rank = number of variables (n=4n = 4), the only solution is the trivial solution.
    • If the rank < number of variables, there are infinitely many solutions.
  3. Compute the rank of the coefficient matrix:

    • Use row-reduction or determinant tests to determine the rank of the matrix.

Reduced Row Echelon Form:

After performing Gaussian elimination, the matrix becomes:

1 & 0 & \frac{31}{19} & \frac{24}{19} \\ 0 & 1 & \frac{21}{19} & \frac{10}{19} \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ ### Analysis: - The rank of the matrix is $$3$$ (number of non-zero rows in RREF). - The number of variables is $$4$$. - Since the rank < number of variables, the system has **infinitely many solutions**. ### Conclusion: The system has **infinitely many solutions**.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Homogeneous Systems
Matrix Rank

Formulas

Ax = 0 (homogeneous system of equations)
Rank of a matrix (number of non-zero rows in RREF)

Theorems

Fundamental Theorem of Linear Algebra
Homogeneous System Properties (trivial and non-trivial solutions)

Suitable Grade Level

Undergraduate Mathematics or Advanced High School