====== Chapter 1: Systems of linear equations ======

===== Linear equations =====

==== First example: a linear equation in two variables ====

Consider the equation \[ 2x+5y=7.\]
This is an equation in two //[[variables]]//, or //[[indeterminates]]//, $x$ and $y$.

A [[solution]] of this equation is a pair of numbers $(a,b)\in \mathbb{R}^2$ so that if we replace $x$ with $a$ and replace $y$ with $b$, then the equation becomes true.

In other words, so that $2a+5b$ really is equal to $7$.

  * $(3,1)$ is not a solution, because $2\times 3+5\times 1\ne 7$
  * $(1,1)$ is a solution, because $2\times 1+5\times 1=7$
  * Other solutions include $(0,\tfrac 75)$, $(0.5,1.2)$, $(6,-1)$, $(3.5,0)$, $(-\tfrac32,2)$, ...

We can't make a complete list of all solutions, since there are //infinitely many// solutions in $\mathbb{R}^2$. However, we can draw the set of all solutions as a subset of $\mathbb{R}^2$. This turns out to be a straight line:
{{ ::g1.jpg-000.jpg?nolink&600 |}}

We say that the equation $2x+5y=7$ is a [[linear equation in two variables]].


==== Definition ====
{{page>linear equation in two variables}}

==== More examples of linear equations in two variables ====

  * $y-x=1$ {{ ::s-000.jpg?nolink&300 |}}
  * $x-y=0$ {{ ::r-000.jpg?nolink&300 |}}
  * $x=0\iff 1x+0y=0$ {{ ::r-001.jpg?nolink&300 |}}