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Example 1

Find all the solutions to the system \(\begin{array}{r} x_1 - 2x_2 + x_3 = 1 \\ x_1 - x_2 - x_3 = 0 \end{array}\) defined over \(\mathbb{R}\).

Example 2

For what values of \(a\) does the system \(\begin{array}{r} x - y + z = 1 \\ -x - y - 3z = 0 \\ y + z = a \end{array}\) defined over \(\mathbb{R}\) have no solution?

Example 3

Find all the solutions to the system \( \begin{split} z - 2w & = 2i \\ z + iw & = 3+i \end{split} \) defined over \(\mathbb{C}\).

Example 4

Let \(A = \begin{bmatrix} 2 & -1 & 3 \\ 4 & 5 & 0\end{bmatrix}\), \(x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\), \(b = \begin{bmatrix} 8 \\ 9\end{bmatrix}\). Write out the system \(Ax = b\) in full.

Example 5

Show that one can transform the system \(\begin{array}{r} x - 2y = 3 \\ x + y = 0 \end{array}\) to the system \(\begin{array}{r} y = -1 \\ x - 2y = 3 \end{array}\) using elementary operations.

Example 6

Let \(x = 2t -1\) and \(y = -t + 1\). For what values of \(t\) is the pair \(x,y\) a solution to the system \(\begin{array}{r} x + y = 1 \\ 2x - y = 2 \end{array}\)?

Example 7

Let \(a,b,c,d,u,v \in \mathbb{R}\). Assuming that \(a \neq 0\) and \(ad - bc \neq 0\), solve the following system for \(x\) and \(y\) using the method of substitution. \begin{eqnarray*} ax + by & = & u \\ cx + dy & = & v \end{eqnarray*}