Simplification with algebraic rules

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

$\begin{array}{rcl} 1\cdot \blue{a} &=& \blue a \\ \\ -1\cdot \blue{a} &=& -\blue a \\ \\ 0\cdot \blue{a} &=& 0 \\ \end{array}$

Requirements to variables

In a rule, a variable represents a random number, but it can be possible to set some requirements for these numbers. For example, the requirement for it to be an integer. If no requirements are mentioned, all numbers can be entered as long as the expressions in the rule make sense mathematically.

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

$4x^2-4x^2 = 0\cdot \blue{4x^2} = 0$

Simplify $-9 x + 7 x y + 6 x -7 x y$ .

$-3 x$

$\begin{array}{rcl} -9 x + 7 x y + 6 x -7 x y &=& -9x + 6 x +7 x y -7xy \\ && \qquad\blue{\text{rule $a+b=b+a$}}\\ &=&-3 x + 7 x y -7 x y \\ && \qquad\blue{\text{coefficients of $x$ added}}\\ &=& -3 x + 0 x y \\ && \qquad\blue{\text{coefficients of $xy$ added}}\\ &=& -3 x + 0 \\ && \qquad\blue{\text{rule $0\cdot a = 0$}}\\ &=& -3 x \\ && \qquad\blue{\text{rule $a+0= a$}}\\ \end{array}$

New example