Question

Solve the system using the inverse of the coefficient matrix:

\begin{array}{l}
x + 4y + 4z = -26 \\
x + 3y + 4z = -22 \\
x + 4y + 3z = -21
\end{array}
\right. \]
The solution is \(x = \), \(y = \), and \(z = \).
Please provide the values for \(x\), \(y\), and \(z\) in the solution to this system of linear equations.

Answer 1

To solve the given system of equations, we form a matrix equation, calculate the inverse of the coefficient matrix, and multiply it with the constants vector to find the variables x, y, and z. The solution to the system is x = 2, y = 2, z = -6.

Step-by-step explanation:

To solve the system using the inverse of the coefficient matrix, we first write the system as a matrix equation Ax = b, where A is the coefficient matrix, x is the column vector of variables, and b is the column vector of constants. The system is:

x + 4y + 4z = -26
x + 3y + 4z = -22
x + 4y + 3z = -21
Which gives us:A =
╔ 1 4 4 ╗
║ 1 3 4 ║
║ 1 4 3 ║
╝
x =
╔ x ╗
║ y ║
║ z ║
╝
b =
╔-26 ╗
║-22 ║
║-21 ║
╝
The next step is to find A-1, the inverse of the coefficient matrix. Then, we multiply A-1 by b to find xA-1b = x
After calculating, the solution to the system of linear equations is:

x = 2
y = 2
z = -6

This process involves using a variety of algebraic techniques including matrix inversion and matrix multiplication. It is important to double-check each step for accuracy.