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--- lecture_1 [2016/01/22 18:40] – rupert
+++ lecture_1 [2016/01/22 19:13] (current) – old revision restored (2015/01/22 12:02) rupert
@@ Line 1: / Line 1: @@
-~~REVEAL~~
 ====== Chapter 1: Systems of linear equations ======
@@ Line 6: / Line 5: @@
 ==== First example: a linear equation in two variables ====
-Consider the equation $ 2x+5y=7$.
+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.
+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 $7$.
-==== $2x+5y=7$ ====
+In other words, so that $2a+5b$ really is equal to $7$.
   * $(3,1)$ is not a solution, because $2\times 3+5\times 1\ne 7$
@@ Line 18: / Line 16: @@
   * 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 $\mathbb{R}^2$.
+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:
-  * 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]].
+We say that the equation $2x+5y=7$ is a [[linear equation in two variables]].