The rules for manipulating numbers (see arithmetic) also apply to algebraic symbols. The only difference is that some of the notation is slightly different. For example, for variables $a$ and $b$, we typically write $a\div b$ as $\frac{a}{b}$ and $a\times b$ as $ab$.

Note: When we multiply variables, together, we typically write variables in alphabetic order. For example, we would write $a\times m \times b$ as $abm$.

Worked Exercises

Example 1

Write the following algebraic expressions in their simplest forms and find the numerical value of each when $a=2$, $b=3$ and $c=1$:

a) $-10a+5a+a$

b) $25c+10c-bc$

Solution

a) $-10a+5a+a=-4a$. When $a=2$ this is equal to $-4\times 2=-8$.

b) $25c+10c-bc=35c-bc$. When $b=3$ and $c=1$ this is equal to $35\times 1-3\times 1=32$.

Note: We could simplify the expression in part b) further by factorising it: \[35c-bc=c(35-b)\]

Example 2

Rewrite the following without using the $\times$ and $\div$ signs in their simplest form: a) $-a\times b+4b-3b\times a$ b) $f\div cf$ c) $-b\div a\times 2m$