Algebra: Variables
Simplification with algebraic rules
When adding numbers, we can exchange the order of how we do it:
\[\blue{\text{first number}} + \green{\text{second number}}= \green{\text{second number}} +\blue{\text{first number}}\]
We can write this in a shorter manner by using variables:
Here #\blue a# and #\green b# represent random numbers. Whichever number you enter for #\blue a# and #\green b#, this equality always holds. For \(\green a\) and \(\blue b\) we can just as well enter other variables, which is convenient when simplifying expressions, like in the example on the right hand side.
Examples
\[{\begin{array}{rcl}{x+\blue{y}+\green{2x} +y }&{=}& {x + \green{2x}
+ \blue{y} +y}\\
&{=}&{3x + y +y} \\
&{=}&{3x+2y}
\end{array}}\]
In mathematics there are a lot of theorems that hold for all numbers. Another example is \[\blue{a} + 0 = \blue a\] This is called an algebraic rule.
On the right hand side are examples of other algebraic rules that hold for every number #\blue{a}#.
Examples
\[\begin{array}{rcl}
1\cdot \blue{a} &=& \blue a \\ \\
-1\cdot \blue{a} &=& -\blue a \\ \\
0\cdot \blue{a} &=& 0 \\
\end{array}\]
From now on we will write down algebraic rules in short, with examples on how to use the rule on the right hand side. With colors we will highlight how the variables are replaced in the rule.
\[1\cdot \blue{a} = \blue a\]
Example
\[3x-2x = 1\cdot \blue{x} = \blue{x}\]
\[-1\cdot \blue{a} = -\blue a\]
Example
\[4x-5x = -1\cdot \blue{x} = -\blue{x}\]
\[0\cdot \blue{a} = 0\]
Example
\[4x^2-4x^2 = 0\cdot \blue{4x^2} = 0\]
#-x#
#\begin{array}{rcl}
5 x + 9 x y -6 x -9 x y &=& 5x -6 x +9 x y -9xy \\
&& \qquad\blue{\text{rule \(a+b=b+a\)}}\\
&=&-x + 9 x y -9 x y \\
&& \qquad\blue{\text{coefficients of \(x\) added}}\\
&=& -x + 0 x y \\
&& \qquad\blue{\text{coefficients of \(xy\) added}}\\
&=& -x + 0
\\ && \qquad\blue{\text{rule \(0\cdot a = 0\)}}\\
&=& -x
\\ && \qquad\blue{\text{rule \(a+0= a\)}}\\
\end{array}#
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