6. [8.58/11.11 Points) DITAILS MY NOTES TANFIN12 2.3.004. Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system as your variables, each representing the columns in turn.) \[ \left[\begin{array}{lll|l} 1 & 0 & 0 & 5 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right] \] (a) Determine whether the system has a solution. The system has one solution. The system has infinitely many solutions. The system has no solution. (b) Find the solution or solutions to the system, if they exist. (If there is no solution, enter NO S your answer in terms of the parameters \( t \) and/or s.) \[ (x, y, z)=(\square) \]
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This row-reduced augmented matrix represents a system of linear equations. The first two rows indicate that the variables \(x\) and \(y\) can be solved directly, while the third row being all zeros suggests that there is no constraint on \(z\). Since there are no contradictory equations, this system has infinitely many solutions. For part (b), you can represent \(z\) as a parameter, let’s say \(z = t\). Therefore, the solution can be expressed as: \[ (x, y, z) = (5, 3, t) \] where \(t\) can take any real number value, providing multiple solutions as \(z\) varies.
