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. This equality always holds whatever number you enter for #\blue a# and #\green b#. 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 of how to use the rule on the right-hand side. With colours, 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
\[4x^2-4x^2 = 0\cdot \blue{4x^2} = 0\]
#-13 x#
#\begin{array}{rcl}
-7 x -4 x y -6 x + 4 x y &=& -7x -6 x -4 x y + 4xy \\
&& \qquad\blue{\text{rule \(a+b=b+a\)}}\\
&=&-13 x -4 x y + 4 x y \\
&& \qquad\blue{\text{coefficients of \(x\) added}}\\
&=& -13 x + 0 x y \\
&& \qquad\blue{\text{coefficients of \(xy\) added}}\\
&=& -13 x + 0
\\ && \qquad\blue{\text{rule \(0\cdot a = 0\)}}\\
&=& -13 x
\\ && \qquad\blue{\text{rule \(a+0= a\)}}\\
\end{array}#
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