## 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*}