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lecture_1
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Chapter 1: Linear systems of equations
Linear equations
Consider the equation \[ 2x+5y=7.\] This is an equation in two variables, or indeterminates, $x$ and $y$.
A solution of this equation will be 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. This turns out to be a line:
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