# Solution sets of linear systems

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##### Intros

###### Lessons

**Solution Set of Linear Systems Overview:**__Homogeneous Systems__

• $Ax=0$

• Trivial, non-Trivial, and general solutions__Solution Sets of Homogeneous Systems__

• Parametric Vector Equation with 1 free variable: $x=tv$

• Parametric Vector Equation with 2 free variable $x=su+tv$

• Parametric Vector Equation with more than 2 free variables__NonHomogeneous Systems__

• $Ax=b$

• General Solutions with an extra vector__Solution Sets of Nonhomogeneous Systems__

• Parametric Vector Equation with 1 free variable: : $x=p+tv$

• Parametric Vector Equation with 2 free variables: $x=p+su+tv$

• Parametric Vector Equation with more than 2 free variables__Difference Between Homogeneous and Nonhomogeneous__

• Extra $p=$ Translation

##### Examples

###### Lessons

**Solution Sets of Homogeneous Systems**

Find the solution set of the homogeneous system in parametric vector form:

$x_1+3x_2+3x_3=0$ $-x_1-3x_2-3x_3=0$ $2x_2+2x_3=0$ - Find the solution set $Ax=0$ in parametric vector form if:

**Solution Sets of Non-homogeneous Systems**

Find the solution set of the nonhomogeneous system in parametric vector form:

$x_1+2x_2-3x_3=2$ $2x_1+x_2-3x_3=4$ $-x_1+x_2=-2$ **Comparing Homogeneous and Nonhomogeneous Systems**

Describe and compare the solution sets of $2x_1+3x_2-5x_3=0$ and $2x_1+3x_2-5x_3=4$- Describe and compare the solution sets of $x_1-4x_2+2x_3=0$ and $x_1-4x_2+2x_3=-3$
**Parametric equation of a line and Translation**

Find the parametric equation of the line through parallel to . Draw all the vectors and the line to show you obtained the line itself geometrically.