Row Operations

Types of systems:

2D ${\mathbb{R}}^2$::

• Non-parallel lines
• Identical lines
• Parallel lines

3D ${\mathbb{R}}^3$:

• Unique solution
• Infinite number of solutions
• No solutions

Row operations:

• (Replacement/ Addition) Add a multiple of one row to another.
• (Interchange) Interchange two rows.
• (Scaling) Multiply a row by a non-zero scalar.

$$\begin{array}{c}\{\begin{array}{l}{x}_1-2x_2+x_3=0\\ 2x_2-8x_3=8\\ 5x_1-5x_3=10\end{array}\end{array}$$

Row₁ +Row ₂

$$\begin{array}{c}\{\begin{array}{l}{x}_1-7x_3=8\\ 2x_2-8x_3=8\\ 5x_1-5x_3=10\end{array}\end{array}$$

Row₂ /2

$$\begin{array}{c}\{\begin{array}{l}{x}_1-7x_3=8\\ x_2-4x_3=4\\ 5x_1-5x_3=10\end{array}\end{array}$$

Row₃ - 5 Row₁

$$\begin{array}{c}\{\begin{array}{l}{x}_1-7x_3=8\\ x_2-4x_3=4\\ 10x_2-10x_3=10\end{array}\end{array}$$

**Note: ${\text{Row}}_1$ represents the original ${\text{Row}}_1$ not the modified ${\text{Row}}_1$

Row₃ - 10 Row₂

$$\begin{array}{c}\{\begin{array}{l}{x}_1-7x_3=8\\ x_2-4x_3=4\\ 30x_3=-30\end{array}\end{array}$$

$x_3=-1$

Substiution

$$x_1=1$$ $$x_2=0$$