ALGEBRA, EQUATIONS AND FORMULAE. INTRODUCTION Algebra essentially involves the substitution of...

ALGEBRA, EQUATIONS AND FORMULAE

INTRODUCTION

Algebra essentially involves the substitution of letters for numbers in calculations, so that we can establish rules and procedures for carrying out mathematical operations which can be applied whatever the actual numbers involved

An equation is simply a statement of equality between two mathematical expressions, and is generally used to enable one or more unknown quantities to be worked out.

A formula is one particular form of generalized mathematical statement, usually expressed in the form of an equation, that sets out a rule which may be applied in particular situations. It enables us to work out an unknown quantity or value provided we know certain specific quantities or values

Objectives

Outline the basic principles of algebra Apply the basic arithmetic operations of addition,

subtraction, multiplication and division to algebraic notation, and simplify algebraic expressions by the process of collecting like terms

Define equations and formulae and outline their uses Find unknown quantities and values from simple

equations by using transposition Outline the principles of formulae and how they are

constructed Rearrange the terms in a formula to isolate different

unknowns.

Algebra

Algebra is a branch of mathematics in which, instead of using numbers, we use letters to represent numbers.

Suppose you have a piece of wood which is 7 meters long and from it you wish to cut a piece 4 meters long. The length of the remaining piece is 3

To find the area of a floor measuring 10 m long and 9 m wide

The distance travelled by a train in 3 hours at a speed of 60 miles per hour

Equations

An equation is simply a mathematical statement that one expression is equal to another. So, for example,

A certain number is added to 4 and the result is 20.

A certain number is multiplied by 4 and the result is 20.

If 4 is taken from a certain number the result is 5.

If a certain number is divided by 3 the result is 1.he statement that "2 +2 = 4" is an equation

Formulae

A mathematical formula (plural "formulae") is a special type of equation which can be used for solving a particular problem.

a formula is an equation which always applies to a particular mathematical problem, whatever the actual values.

ALGEBRAIC NOTATION

As algebraic letters simply represent numbers, the operations of addition, subtraction, multiplication and division are still applicable in the same way. However, in algebra it is not always necessary to write the multiplication sign. So, instead of "a x b", we would write simply "ab" (or sometimes "a.b", using the full stop to represent multiplication).

Addition and Subtraction

Multiplication

Division

We can always cancel like terms in the numerator and denominator

Indices, Powers and Roots

You will remember that indices are found when we multiply the same number by itself several times. The same rules apply in algebra

Multiplication of indices

Division of indices

Showing reciprocals

Negatives denote reciprocals

Raising one power by another

Roots

Roots of powers

remember that the numerator of a fractional index denotes a power and that the denominator denotes a root.

Collecting like terms

Brackets in Algebra

The rules for brackets are exactly the same in algebra as in arithmetic. However, if we have unlike terms inside the brackets, it is not possible to collect them together before removing the brackets.

SOLVING EQUATIONS

When we talk about solving an equation, we mean finding the value of the unknown or unknowns using the other numbers in the equation

What you do to one side of the equation, you must also do to the other

Another way of thinking about this is that an equation is like a balance. If the weights of a balance are equal on both sides it is "in balance". You can add an equal weight to each side, or take an equal weight from each side, and it will remain "in balance".

(a) The same number may be added to both sides of the equation

(b) The same number may be subtracted from both sides of the equation

(c) Both sides of the equation may be multiplied by the same number

(d) Both sides of the equation may be divided by the same number.

Transposition

Transposition is a process of transferring a quantity from one side of an equation to another by changing its sign of operation. This is done so as to isolate an unknown quantity on one side

A multiplier may be transposed from one side of an equation by changing it to the divisor on the other, Similarly, a divisor may be transposed from one side of an equation by changing it to the multiplier on the other.

Equations with the Unknown Quantity on Both Sides

These equations are treated in the same way as the other equations we have met so far. We simply keep transposing terms as necessary until we have collected all the unknown terms on one side of the equation

FORMULAE

A formula is a mathematical model of a real situation.

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