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Chapter 1: Systems of linear equations

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Linear equations

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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 $7$.

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$(3,1)$ 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 $(0,\tfrac 75)$, $(0.5,1.2)$, $(6,-1)$, $(3.5,0)$, $(-\tfrac32,2)$, …

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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
$2x+5y=7$ is a linear equation in two variables.

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If $a,b,c$ are any fixed numbers, then equation \[ ax+by=c\] is a linear equation in two variables.

When you draw the set of all solutions of a linear equation in two variables, you always get a straight line in the $x$-$y$ plane.

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