02Apronumerals

= Pronumerals = toc A pronumeral is a letter (or sometimes a symbol or a greek letter) that we use in place of a number.

Pronumerals are sometimes called ** variables ** {because the number they represent can vary}.

{The word __**pronumeral**__ means "in place of a number(numeral)" in the same way that pronoun means "in place of a noun."}

Literacy
A ** coefficient ** is a number in front of a pronumeral. It tells you how many lots of that pronumeral is involved.
 * ** Eg : ** 3a .. ==> The coefficient is 3. ... There are 3 lots of a. ... (3 × a)
 * **Eg :** –2xy .. ==> The coefficient is –2. ... There are –2 lots of xy. ... (–2 × xy)

If no number is visible in front of a pronumeral, the coefficient is 1.
 * ** Eg : ** m 2 .. ==> The coefficient is 1. ... There is 1 lot of m 2 ... (1 × m 2 )

A ** term ** is the product of one or more pronumerals, together with a coefficient, that appear as one "group". **Eg :** The following are all terms:
 * 3a
 * m 2
 * –2xy
 * 6ab 2
 * 7

Note: The standard way to write a term with several pronumerals is with the pronumerals in alphabetical order.
 * ** Eg : ** 7ab is preferred to 7ba {but 7ba is not wrong}

If the term has no pronumeral (eg 7), it is called a ** constant term ** or sometimes just a ** constant **.

An ** expression ** is one or more terms joined by "+" or "–".
 * Eg : ** The following are all expressions:
 * 7a + 2b + 5c
 * m 2 – 3m
 * 6ab
 * 7

An ** equation ** has one equal sign with an expression on each side of the "=". **Eg :** The following are all equations:
 * 3a + 2 = 7
 * m 2 – 2m = 5m + 3
 * a = 3
 * x 2 + 5x + 9 = 0

Like Terms
Algebra terms can only be __added__ or __subtracted__ if they are **__like terms__**.


 * Like terms ** have exactly the same letter part. {the coefficient doesn't matter}

Note : The letters can be arranged in a different order as long as they are all there.

The following pairs are like terms
 * 3a //and// 5a
 * 3xy //and// –2yx
 * m 2 //and// 17m 2
 * 10 //and// 6
 * 5xy 2 //and// 7y 2 x

Adding & Subtracting
Only __like terms__ can be __added__ or __subtracted__.

Simply add or subtract the coefficients and keep the pronumerals the same.

If there are terms that are not like terms, they remain as an expression seperated by "+" or "–".

Note : The "–" sign always belongs to the term that comes __after__ it.
 * ** Eg : ** The expression : 3x – 5y ==> has 2 terms : +3x and –5y

** Example 1 **
math \\ \textbf{(a)} \quad 3x + 5x = 8x \qquad. \\ . \\ \textbf{(b)} \quad 7ab - 2ba = 5ab \qquad. \\ . \\ \textbf{(c)} \quad 5a + 3b - a + 2b = 4a + 5b \qquad. math

Multiplying
Any algebra terms can be multiplied (they don't have to be like terms)
 * multiply the coefficients together (like normal numbers)
 * push the letters together
 * mutliples of the same pronumeral should be written in index form

** Example 2 **
math \\ \textbf{(a)} \quad 4a \times 3b = 12ab \qquad. \\ . \\ \textbf{(b)} \quad 5m \times 2m = 10m^2 \qquad. \\ . \\ \textbf{(c)} \quad ab \times 5a = 5a^2b \qquad. math

Dividing

 * Write the division as a fraction.
 * Treat the coefficients as a normal fraction and simplify the fraction if possible
 * leave the fraction part as improper -- don't change to a mixed number
 * If a pronumeral appears in the numerator and the denominator, it can be cancelled.

** Example 3 **
math \\ \textbf{(a)} \quad \dfrac{9a}{6b} = \dfrac{3a}{2b} \qquad \qquad \textit{cancel 3 from top and bottom} \qquad. \\ . \\ . \\ \textbf{(b)} \quad \dfrac{4ab}{3a} = \dfrac{4b}{3} \qquad \qquad \textit{cancel a from top and bottom} \qquad. \\ . \\ . \\ \textbf{(c)} \quad \dfrac{2a}{6ab} = \dfrac{1}{3b} \qquad \qquad \textit{cancel 2 and a} \qquad. \\ . \\ . \\ \textbf{(d)} \quad \dfrac{3m^2}{m} = \dfrac{3m}{1} = 3m \qquad \textit{cancel m, then write as a whole number} math

Note: math \text{In example } \textbf{(3d)} \text{ above : } \dfrac{3m^2}{m} \text{ can be written as } \dfrac{3mm}{m} = \dfrac{3m}{1} \qquad. math

... ... ... __One__ of the "m"s from the numerator cancels with the "m" from the denominator, leaving "3m" in the numerator.


 * Example 3 ** (cont)

math \textbf{(e)} \quad \dfrac{18x^2yz}{10xyz^2} = \dfrac{9x}{5z} \qquad \qquad \textit{cancel 2 from the numbers then x and y and z} \qquad. math

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