Systems of Linear Equations

Systems of Linear Equations: Algebra

Introduction

Line, Plane, Space, Etc.

Pictures of Solution Sets

Parametric Description of Solution Sets

Which of the following equations is linear in the unknowns $x,y,z$?

How many solutions does the system

have?

In one word: a system of linear equations that has no solutions at all is called ______ .

The plane $x + y + z = 1$ in $\mathbb{R}^3$ has the parametric form $(x,y,z) = (1-t-w,\,t,\,w)$. How many parameters (free variables) does that description use? Enter a number.

The two equations $x + y + z = 1$ and $x - z = 0$ describe a line in $\mathbb{R}^3$. Solve them using $z$ as the free variable, writing the line as $(x,y,z) = (t,\ ?,\ t)$. Enter the $y$-coordinate as an expression in $t$ (for example: 1-2t).

In your own words, state the single most important definition or theorem of this section, and explain what it says.