Questions: Solve the following system of equations. 4x - 2y + 2z = 20 2x - 7y + 2z = 25 6x + 15y - 4z = -3

Transcript text: Solve the following system of equations. \[ \left\{\begin{array}{l} 4 x-2 y+2 z=20 \\ 2 x-7 y+2 z=25 \\ 6 x+15 y-4 z=-3 \end{array}\right. \]

Solution

Solution Steps

Step 1: Formulate the System of Equations

We are given the following system of equations: \[ \begin{align_} 4x - 2y + 2z &= 20 \quad (1) \\ 2x - 7y + 2z &= 25 \quad (2) \\ 6x + 15y - 4z &= -3 \quad (3) \end{align_} \]

Step 2: Represent the System in Matrix Form

The system can be represented in matrix form as \( Ax = b \), where: \[ A = \begin{bmatrix} 4 & -2 & 2 \\ 2 & -7 & 2 \\ 6 & 15 & -4 \end{bmatrix}, \quad x = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad b = \begin{bmatrix} 20 \\ 25 \\ -3 \end{bmatrix} \]

Step 3: Perform Gaussian Elimination

Through Gaussian elimination, we transform the augmented matrix \([A | b]\) step by step until we reach the reduced row echelon form: \[ \begin{bmatrix} 1 & 0 & 0 & 5 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & -3 \end{bmatrix} \]

Step 4: Extract the Solutions

From the final reduced row echelon form, we can directly read the solutions for the variables: \[ \begin{align_} x &= 5 \\ y &= -3 \\ z &= -3 \end{align_} \]