Questions: Systems of Equations and Matrices Writing solutions to 3 × 3 systems of linear equations from augmented... 0 augmented matrices for two linear systems in the variables x, y, and z are given b augmented matrices are in reduced row-echelon form. - each system, choose the best description of its solution. applicable, give the solution. (a) [ 1 0 -1 2 0 1 1 5 0 0 0 0 ] The system has no solution. The system has a unique solution. The system has infinitely many solutions. (x, y, z)=(x, , ) (x, y, z)=(, y, ) (x, y, z)=(π, , z)

Solution Steps

To solve the given system of equations represented by the augmented matrix, we need to interpret the reduced row-echelon form (RREF) of the matrix. The RREF indicates the relationships between the variables. In this case, the third row is all zeros, which suggests that the system has infinitely many solutions. We can express the solutions in terms of a free variable, typically denoted as \( z \).

1. From the first row, we have the equation \( x - z = 2 \), which can be rearranged to \( x = z + 2 \).
2. From the second row, we have the equation \( y + z = 5 \), which can be rearranged to \( y = 5 - z \).
3. The third row does not provide any new information.

Thus, the solution can be expressed in terms of the free variable \( z \).

The given augmented matrix in reduced row-echelon form is: \[ \left[\begin{array}{ccc:c} 1 & 0 & -1 & 2 \\ 0 & 1 & 1 & 5 \\ 0 & 0 & 0 & 0 \end{array}\right] \] This corresponds to the following system of equations:

1. \( x - z = 2 \)
2. \( y + z = 5 \)

From the first equation, we can express \( x \) in terms of \( z \): \[ x = z + 2 \] From the second equation, we can express \( y \) in terms of \( z \): \[ y = 5 - z \]

Since \( z \) is a free variable, we can express the solution set as: \[ (x, y, z) = (z + 2, 5 - z, z) \] This indicates that there are infinitely many solutions depending on the value of \( z \).

Final Answer