Simplification

Algebra: Variables

Simplification

SimplificationFor #4 \cdot 5 x# we can also write #20x#.
Hence, we can write #4 \cdot 5 x# in a more simplified manner. We can call this simplifying an expression.

Example

\[\begin{array}{rcl}
{\blue{5\cdot 8} x} &{=}& {\blue{40}x}
\end{array}\]

Simplification of multiplications

The product #\blue x\cdot \green y# is the same as #\green y\cdot \blue x#.
We can exchange the order when multiplying. In general, we choose to have the #\purple{\text{number}}# up front and the letters in the order of the alphabet.

Example

\[\begin{array}{rcl} {3 \cdot \green{y} \cdot 6 \cdot \blue{x}} &{=}&{3 \cdot {6} \cdot {\blue{x}} \cdot {\green{y}}} \\&{=}&{{\purple{18}} {\blue{x}} {\green{y}}} \end{array}\]

More examples

On the right-hand side, we see a different example in which we can exchange the order in the multiplication in such a way that the number is up front and the letters are in the order of the alphabet.

Example

\[\begin{array}{rcl}
5\cdot \blue{x}\cdot -3\cdot \green{y} &=& 5\cdot -3 \cdot \blue{x}\cdot \green{y} \\
&=& \purple{-15}\blue{x}\green{y}
\end{array}\]

Simplification of additions and subtractions

The sum #3\blue{x} + 2\blue{x}# has similar terms.
Within similar terms, we find exactly the same variables with the same exponents.

Similar terms can be simplified by combining similar terms together. To combine like terms, we add the coefficients.

Equations with numbers

Adding similar terms can be compared to the process of repeatedly adding numbers.

Numbers	Variables
\(\begin{array}{rcl} 2 \cdot \blue{4}+ 3\cdot \blue{4} &=& \blue 4+\blue 4+\blue 4+\blue 4+\blue 4 \\ &=& 5 \cdot\blue{4}\end{array}\)	\(\begin{array}{rcl} 2 \blue{x}+ 3 \blue{x} &=& \blue x+\blue x+\blue x+\blue x+\blue x \\ &=& 5\blue{x}\end{array}\)

Simplify #10x^2+5x^2#.

#15x^2#

#\begin{array}{rcl}
10x^2+5x^2 &= &15x^2\\
&&\blue{\text{coefficients \(10\) and \(5\) of \(x^2\) added}}
\end{array}#

New example