Math Problem Statement

Solve the following system of homogeneous linear equations: 1. 2x + 2y + 4z = 0 2. w - y - 3z = 0 3. 2w + 3x + y + z = 0 4. -2w + x + 3y - 2z = 0

Solution

You have uploaded an image with a problem asking to solve a system of homogeneous linear equations. The equations provided are:

  1. 2x+2y+4z=02x + 2y + 4z = 0
  2. w−y−3z=0w - y - 3z = 0
  3. 2w+3x+y+z=02w + 3x + y + z = 0
  4. −2w+x+3y−2z=0-2w + x + 3y - 2z = 0

This is a system of 4 equations in 4 variables: ww, xx, yy, and zz. Since it's a homogeneous system, one solution is always the trivial solution where all variables are zero: w=x=y=z=0w = x = y = z = 0. However, we will look for non-trivial solutions if they exist.

To solve this system, we can represent it as a matrix and apply Gaussian elimination or similar techniques to find the solutions.

Let me proceed with solving this system step-by-step.The system of equations, represented in reduced row echelon form (RREF), is:

1 & 0 & -1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ From this matrix, we can derive the following equations: 1. $$w - y = 0$$, which implies $$w = y$$ 2. $$x + y = 0$$, which implies $$x = -y$$ 3. $$z = 0$$ Since the last row is entirely zeros, the system is consistent, and $$y$$ is a free variable. We can express the solution in terms of $$y$$: - $$w = y$$ - $$x = -y$$ - $$z = 0$$ Thus, the general solution is: $$w = y, \quad x = -y, \quad z = 0$$ where $$y$$ can be any real number. Would you like more details or have any questions about the solution? Here are five related questions you might explore: 1. How can you verify the solution by substitution? 2. What is the significance of the free variable in the solution of this homogeneous system? 3. How does Gaussian elimination help solve systems of linear equations? 4. Can you solve the same system using matrix inversion? 5. What happens if the system had no free variables? **Tip**: For homogeneous systems, a non-trivial solution exists only if the determinant of the coefficient matrix is zero.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Homogeneous Systems of Linear Equations
Gaussian Elimination
Matrix Theory

Formulas

Matrix representation of a system of linear equations
Gaussian Elimination process for solving systems
Reduced Row Echelon Form (RREF)

Theorems

The existence of non-trivial solutions for homogeneous systems
The Rank-Nullity Theorem

Suitable Grade Level

Grades 10-12 (High School to Early University)

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