Rules for Algebra _ www.juniorcertsolutions.com

Rule 1 Addition of  two Numbers with the same sign The result is the sum of the numbers with the given sign ( the sign of the two numbers )  .

 When we add two  plus numbers we get a plus number          Ex   :   3+6=9. 

 When we add two minus numbers we get a minus number   Ex :  -5 + - 6 = - 11 .

Example 5 + 6 + 8 = 19  .   Example    -5  +  -  8   +  -  6 =   -  19 .

Rule 2Addition of two numbers with different signs  the result is the difference between the numbers with the sign of the" bigger number " .

Ex  6 + - 9 = - 3 :  Ex  19 + -5 =  14 .  Ex   -15   + 13 = - 2 .

Note  Ex :  14    - 9  is the same as 14  +  -  9  =  5 . and -  7 -  5 is really -7  + - 5 = - 12

Example simplify -  6 +   9    -   7  +  3 +   6 -  8 = -  6  - 7 - 8 + 9 + 3 + 6 = - 21 + 18 = - 3 .

Note we add the minus numbers ,we add the plus numbers then tidy it up .

Rule 3 : Multiplication and signs :

  The basic rules for multiplication and signs  are as follows

(Minus )  x  ( Minus ) =  Plus .   ( Minus )  x  ( Plus )  = Minus ,  (Plus )  x  (Plus )  =  Plus .

This can be expressed as follows that (a) when we multiply two  numbers with the same sign   we get plus   ( b) when we multiply  two numbers with difference signs  we get a minus .

Example :  - 7  x  - 5 =  + 35  ,   -7  x  5  = -35  .  Example  - 7. x  -  4  x  3 =  28  x  3 =  84 .

Example : - 5 x - 6 x - 2 =  30 . x. - 2 = - 60 . Note if we are asked to multiply three or more numbers remember you  can only  multiply two numbers at a time .

Example   - 4  x -3 x 5 x -2 =  12  x 5 x - 2 = 60  . x - 2    =  - 120 .

Rule 4 :  Letters

Letters are often used in place of numbers , letters are often referred to  as variables or unknowns . Rules 1,2,3 above apply to all calculations with letters , we have to remember one further rule which applies to letters that is we can only add things together that are the same  ie we can only add x's to x's and y's to y's etc  .  Note also 5a means 5 times a .

Example 5a - 7a = - 2a , Example  - 3x +  7x  = 4x, Example 5a + 6b - 4a - 11b ,= 5a - 4a  + 6b - 11b =  a - 5b .

Multiplying letters by letters : 

Rule 5 :    Brackets :

(a) A number immediatly outside a bracket means that everything inside the bracket gets multiplied by that number   Ex

                            Ex

(b)Minus sign  outside a bracket will change all the signs inside the brackets when the brackets are removed : ex

(c)Minus number outside a bracket does two things (i) everything inside the bracket gets multiplied by the number and (ii) all the signs change

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Brackets by brackets :

Multiply everything in the second bracket by everything in the first bracket and tidy it up .               

Mathematical Words :

The following words appear frequently  in Mathematics Exam Papers .

(1Expression :

This is a collection of letters and numbers : eg  (4x + 3y)(5x-7y)  : 7x2+4xy-6 .

Words associated with expressions.

Simplify : Means get rid of the brackets and tidy it up :

ex Simplify

Factors : The factors of of an expression are two or more things which when  multiplied together give that expression

.Example the factors of 14 are 7 and 2 because 7 x 2 =14

Factorise : means find the factors of : Example factorise

(2) Equation :

This is an expression which contains an equals sign : All equations contain an equals sign .

7x- 5y = 10 :  4x2-6x+5 = 0,  3x+2y = 4

                                               4x- 5y = 10.

Word associated with equations

Solve : Means find the value(s) of the letter that makes the equation true .

Example Solve

Types of equation (1) Linear equation one variable Example 3x-4=10

(2)Linear equations 2 variables  these are called simultaneous equations

(3)Quadratic equations : Equations of the form  are called quadratic equations:

the solutions of a quadratic equation are called the roots of the equation , all quadratic equations have two roots!

 is a quadratic equation

Inequality : This is an expression which contains one of the following symbols

Word associated with Inequalities  Solve;

Example Solve  

Verify : this means show that the information you are given is true

Verify if                            

Calculate  means  work out ( using maths)  the value of : Example Calculate

Evaluate : (normally applies to a given expression) we are asked to simplify and find the value of .

Example Evaluate

Words associated with fractions :

Numerator : the top half of the fraction, Denominator :the bottom half of the fraction.

Common denominator : this is the smallest number that the bottom halves of two or more fractions will divide.

Example the common denominator of  is 30 as it is the smallest number that can be divided by 2,3, and 5 .

Lowest common multiple (LCM) is another name for the smallest number that a given set of numbers will divide .

Highest Common factor (HCF) : this is the biggest factor a set of numbers have in common

example the highest common factor of 15,25 and 30 is 5  : the HCF of 8,12,and 28 is 4 .

Words Associated with graphs :

Plot this means place on a coordinate plane some given points .

Estimate (from your graph )  read answers from your graph .

Types of Numbers and their Symbols

(1) Natural Numbers  Symbol N This is the set  of  positive whole numbers .

 N =(0,1,2,3,4,.....)

On the numbers line we use" dots " to indicate Natural numbers

(2) Integers Symbol Z This is the Set of  positive and negative whole numbers  Z={-3,-2,-10,1,2,3}

on the number line we show the integers as "dots"

(3)Rational Numbers Symbol Q : This is the set of numbers that can be written as a fraction

Most numbers  can be written as a fraction eg

In general you will not be asked to put rational numbers on the number line .

(4)Irrational Numbers Symbol Ir : Numbers that cannot be written as a fraction :

The only numbers that cannot be written as a fraction are numbers such as  (a non repeating non terminating decimal) or surds such as

(5)Real numbers Symbol R : This is the set which includes all of the above types of numbers On the number line we use a thick line to show real numbers

(6)Complex numbers : These are numbers of the form x+yi where x and y are Real numbers and  ( i ) =

complex numbers are plotted on an Argand diagram.

Inequality symbols

 In general read all information from left to right

>" is greater than ",    x is greater than y .  <" is less than ",  a is less than b

 "is less than or equal to"   x is less than or equal to k .: "is greater than or equal to"

  p is greater than or equal to h .

Compound inequalities  read from the middle to the left then from the middle to the right

the above inequality reads " x is greater than or equal to a and x is less than or equal to b "

Symbols and Sets

: "is an element of "belongs to a particular set  is not an element of .

. . Union  Au B means the set of all the elements of A and B with none repeated

 intersection , A  B means the set of all the elements that A and B have in common .

 "is a subset of " :A B  means that A is a subset of B , A contains some or all of the elements of B but no other elements . means  is not a subset .

The cardinal number of a set , this indicates the number of elements in the set .

More Symbols :

 means is perpendicular to : A  B means A and B meet at right angles .

// means is parallel to .  means implies   :  abc means angle abc .  also means angle abc

 means the size of the angle ,  The symbol   in general means the size of or the measure of or the lenght of  ,  but another meaning is the absolute value of  meaning the positive value of something

Function notation : A function is a rule which is shown as an expression in x Ex this states that f is a function which turns x into x + 4 ,(4 is added to every x )  so f(3) = 3+ 4, f(7) = 7 + 4 . The way functions are shown can vary the same function can be shown as follows  The first R is the domain of the function these are the numbers that can be used in the function ,the second R is the Codomain of the function these are the numbers that can come out of the function  . Functions are often referred as maps and are said to Map (connect up ) the elements of the Domain with those of the Codomain . But the most common notation is f(x) equals .