For the row-reduced augmented matrix \(\left(\begin{array}{cc|c}1&0&3\\0&1&-2\end{array}\right)\), state the solution of the system.

In a row-reduced augmented matrix, what does the row \(\left(\begin{array}{ccc|c}0&0&0&5\end{array}\right)\) say about the solution set?

For \(\left(\begin{array}{ccc|c}1&0&-4&2\\0&1&3&-1\\0&0&0&0\end{array}\right)\), identify the pivot variables and the free variable.

A homogeneous system in three variables row-reduces to \(\left(\begin{array}{ccc|c}1&2&0&0\\0&0&1&0\end{array}\right)\). Does it have only the trivial solution or infinitely many solutions?

Describe the solution set of \(\left(\begin{array}{ccc|c}1&0&2&7\\0&1&-5&4\end{array}\right)\).

Solve the system represented by \(\left(\begin{array}{ccc|c}1&-1&0&6\\0&1&2&3\\0&0&1&-2\end{array}\right)\).

For \(\left(\begin{array}{cccc|c}1&0&3&-1&2\\0&1&-2&4&5\\0&0&0&0&0\end{array}\right)\), write the solution set using two parameters.

A system row-reduces to \(\left(\begin{array}{ccc|c}1&0&2&3\\0&1&-1&8\\0&0&0&0\end{array}\right)\). Explain why the system has infinitely many solutions, not one solution.

Solve \(x+2y-z=4\), \(2x+4y-2z=8\), and \(y+z=1\), and describe the solution set.

Row reduction of \((A\mid \mathbf b)\) gives \(\left(\begin{array}{cccc|c}1&0&2&0&-1\\0&1&-3&0&4\\0&0&0&1&6\end{array}\right)\). Write all solutions as \(\mathbf{x}_p+\mathbf{x}_h\).

For the homogeneous system with row-reduced matrix \(\begin{pmatrix}1&-2&0&5\\0&0&1&-3\\0&0&0&0\end{pmatrix}\), find a basis-style parametrisation of the solution set.

Suppose \(\mathbf u\) and \(\mathbf v\) are both solutions of \(A\mathbf x=\mathbf b\). Show that \(\mathbf u-\mathbf v\) solves the associated homogeneous system.

Find all values of \(k\) for which the system \(x+y=2\) and \(2x+2y=k\) is consistent, then describe the solution set for those values.

For what value of \(k\) does \(\left(\begin{array}{ccc|c}1&2&-1&3\\0&1&4&5\\0&0&k-2&6\end{array}\right)\) have infinitely many solutions? Explain.

Classify the solutions by parameter \(a\) for \(x+y+z=1\), \(2x+2y+2z=2\), and \(x+y+az=3\).

For \(\left(\begin{array}{ccc|c}1&1&0&2\\0&a&1&3\\0&0&a-2&4\end{array}\right)\), determine whether each value of \(a\) gives a unique solution, no solution, or infinitely many solutions.

A consistent system in five variables has row-reduced augmented matrix with pivot columns \(1\), \(3\), and \(5\). Explain the dimension of its solution set and write its general form using named vectors.

A student says: "A homogeneous system with a free variable might have no solution if the parameter is chosen badly." Diagnose the error.

Explain why the full solution set of a consistent inhomogeneous system is not usually a vector space, even though its homogeneous part is.

A row-reduced augmented matrix has no inconsistent row and has more columns of variables than pivot columns. Prove that the system has infinitely many solutions.