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Title: Pure Mathematics 02 - Algebra
Description: Contains a full set of notes on an introduction to the principles of algebra. Useful fundamentals for 1st year science/engineering students covers basics of algebra, forming linear equations, quadratic equations and simultaneous equations.
Description: Contains a full set of notes on an introduction to the principles of algebra. Useful fundamentals for 1st year science/engineering students covers basics of algebra, forming linear equations, quadratic equations and simultaneous equations.
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Pure Mathematics 02 β Algebra
01 Basics of Algebra
Pure Mathematics 02- Algebra
01 Basics of Algebra
Simplifying Algebraic Expressions
Algebra is the branch of Mathematics that deals with the manipulation of letters or symbols, that are
being are used to represent numbers
...
Simplifying an algebraic expression will not necessarily result in a single
term
...
'Like
terms' are those whose letter parts are identical
...
It is conventional to put the number first, followed by the letters in alphabetical
order
...
It is important to remember that any term always take the sign that is in front of it
...
Subtracting is equivalent to moving to the left along a number line
...
ο·
ο·
ππ + ππ β ππ = ππ β ππ
ππ + ππ β ππ + π = ππ β ππ + ππ + π = π + ππ
In an algebraic expression it is unusual to see either a multiplication sign (Γ) or a division sign (Γ·)
Remember: π π = π + π But, since multiplication is repeated addition π π = π Γ π is also true
To multiply one expression by another, the numbers are multiplied and the letters are multiplied:
ππ Γ ππ = π Γ π Γ π Γ π = π Γ π Γ π Γ π = π Γ ππ = πππ
When we multiply together two letters that are the same, this results in a power of that letter
...
1
Pure Mathematics 02 β Algebra
01 Basics of Algebra
In arithmetic, the calculation ππ Γ· π can be written as the fractional expression
algebra, π Γ· π is more usually written in the fraction form
π
...
π
Similarly, in
Numerical fractions are simplified by cancelling common factors that occur in both numerator and
denominator
...
The common factors may be
numbers or letters
...
Brackets are also
frequently used in algebra:
π Γ (π + π) means add π to π before multiplying by 2
This is usually written without the multiplication sign: π(π + π)
π Γ (π + π) means (π + π) + (π + π) = π + π + π + π = π Γ π + π Γ π = ππ + ππ
Working this out is called expanding the brackets
...
Note that the number or letters may appear in front of the bracket or after it
...
e
...
Examples - Evaluate each expression if: π = π, π = π, π = π, π = ππ
π + π β π = π + π β π = π)
ππ β ππ = (π Γ π) β (π Γ π) = ππ β ππ = βπ
π π ππ = π π Γ π Γ ππ = πππ)
Integers
The set of natural numbers is 1,2,3,4β¦ The set of integers includes the positive whole numbers and
their negatives, together with the number zero: β¦-3,-2,-1, 0, 1, 2, 3β¦
The rules for combining positive/negative numbers and algebraic terms are summarised in the table:
2
Pure Mathematics 02 β Algebra
Addition
Subtraction
Multiplication
Division
Rule
π + (+π) = π + π
π + (βπ) = π β π
π β (+π) = π β π
π β (βπ) = π + π
(+π) Γ (+π) = ππ
(+π) Γ (βπ) = βππ
(βπ) Γ (+π) = βππ
(βπ) Γ (βπ) = ππ
π
(+π) Γ· (+π) =
π
π
(+π) Γ· (βπ) = β
π
π
(βπ) Γ· (+π) = β
π
π
(βπ) Γ· (βπ) =
π
01 Basics of Algebra
Numerical Example
(βπ) + (+π) = βπ
(βπ) + (βπ) = βπ
(βπ) β (+π) = βπ
(βπ) β (βπ) = βπ
(+π) Γ (+π) = ππ
(+π) Γ (βπ) = βππ
(βπ) Γ (+π) = ππ
(βπ) Γ (βπ) = ππ
(+ππ Γ· (+π) = π
(+ππ) Γ· (βπ) = βπ
(βππ) Γ· (+π) = βπ
(βππ) Γ· (βπ) = π
Addition and Subtraction
Combine signs before attempting to add or subtract
...
ππ + (+ππ) = ππ + ππ = ππ
2
...
βππ + (βππ) = βππ β ππ = βππ
4
...
Examples β Simplify:
1
...
(βππ) Γ (+ππ) = βππππ
3
...
βππ π π
πππ π
=β
π
ππ π
3
Pure Mathematics 02 β Algebra
01 Basics of Algebra
Brackets
The rules for removing brackets preceded by a plus or minus sign are summarised below:
π + (π + π) = π + π + π
π + (π β π) = π + π β π
π β (π + π) = π β π β π
π β (π β π) = π β π + π
Note that the minus sign immediately in front of the bracket changes any signs inside when the
bracket is removed
...
βπ(ππ β ππ) = β(ππ β πππ) = βππ + πππ
2
...
ππ β ππ(π β π) = ππ β (ππ β ππ π ) = ππ β ππ + ππ π = βπ + ππ π
4
...
Examples β Let π = π, π = βπ, π = π, π = βπ
1
...
ππ β ππ = π(βπ) β π(π) = (βππ) β (π) = βππ β π = βππ
πππ β ππ π = π(βπ)(π) β π(βπ π ) = βππ β π(π) = βππ β ππ = βππ
4
Pure Mathematics 02 β Algebra
02 Forming Linear Equations
02 Forming Linear Equations
Many problems are solved more quickly if they are first written in algebraic terms
...
All equations must contain an equals sign
...
If the unknown quantity is π, then a linear equation in π is one that may be reduced to multiple of π
and a pure number term
...
Example: My brother is twice as old as I am and the sum of our ages is 42
...
To solve ππ + π = ππ means finding the value of π which makes the equation true
...
An equation must always be balanced i
...
the left hand side must always equal the right hand side
...
To
maintain balance, 7 must also be deducted from the right hand side giving ππ = ππ
ππ must now be reduced to π by dividing both sides by 3, giving π = π
The stages are:
ππ + π = ππ
Subtract 7 from both sides:
ππ + π β π = ππ β π
Giving:
ππ = ππ
Divide both sides by 3:
ππ
π
Giving:
π= π
=
ππ
π
5
Pure Mathematics 02 β Algebra
02 Forming Linear Equations
Harder Equations
The method used above can be very time consuming and, when solving more difficult equations, a
quicker method is used
...
e
...
Showing the division on the right hand side only we have:
π=
ππ
= π
π
I
...
the 3 has changed sides and changed sign from Γ to Γ·
The rule for eliminating a quantity from one side of an equation is change to the opposite side and
change to the opposite sign
...
6
Pure Mathematics 02 β Algebra
03 Quadratic Equations
03 Quadratic Equations
Quadratic Expressions
A linear expression in terms of the variable π is one that consists of a multiple of π and a constant
term
...
Example:
(π + π)(π + π) = π(π + π) + π(π + π) = π π + ππ + ππ + ππ = π π + ππ + ππ
(ππ β π)(π + π) = ππ(π + π) β π(π + π) = ππ π + ππ β ππ β ππ = ππ π β π β ππ
The general form for a quadratic expression, which can be seen in the examples above, is:
ππ π + ππ + π
Where π, π, π are numbers and π cannot be zero
...
It seems reasonable to believe that this process can be reversed, i
...
a quadratic expression can be
written as a product of two linear factors
...
Not all quadratic expressions can be factorised easily
...
A quadratic equation is of the form:
ππ π + ππ + π = π [π β π]
The variable does not have to be π, but the powers of the variable have to be 2, 1 and 0
...
This concept is very important for the solution of a quadratic equation by factors
...
Therefore either: π = π ππ π = βπ
7
Pure Mathematics 02 β Algebra
03 Quadratic Equations
Note: A quadratic equation always has two possible solutions
...
Therefore either: π = π ππ π = π
So π = π
Solving Quadratic Equations using the Formula
The equation ππ π β ππ β π = π does not have simple factors
...
The standard form of a quadratic equation is:
ππ π + ππ + π
Where π, π, π are numbers and π cannot be zero
...
)
8
Pure Mathematics 02 β Algebra
04 Simultaneous Equations
04 Simultaneous Equations
Eddie has just bought a new pen and his friend Pete would also like one, but Eddie canβt remember
how much it cost
...
20
...
Fiona then remembers that she bought two of the same type of pen and a ruler for Β£3
...
Pete can now work out the cost of the pen:
π + π = πππ
ππ + π = πππ
The extra pen which Fiona bought must have cost (350 - 220) pence, i
...
130 pence
Therefore the pen cost Β£1
...
If an equation has two unknown values (variables), it cannot be solved on its own
...
e
...
Three variables would
require three simultaneous equations etc
...
Example β Solve the equations:
ππ + ππ = ππ (π)
ππ + ππ = ππ (π)
The only difference between the left hand sides of the two equations is that equation (1) has ππ
more, which must be balanced by the extra 6 on the right hand side
...
Substitute π = π into (2):
π Γ π + ππ = ππ
π + ππ = ππ
ππ = ππ
π= π
The solution is π = π, π = π
...
Example:
ππ + ππ = πππ (π)
ππ + ππ = πππ (π)
Begin by deciding which variable to eliminate, π in this case
...
In this case multiply (1) by 3 to give equation (3) and
multiply (2) by 2 to give equation (4):
ππ + ππ = πππ (π)
ππ + ππ = πππ (π)
Subtract (4) from (3):
π = ππ
The value of π can be found by substituting the value of π into one of the original equations, (1) or
(2)
Title: Pure Mathematics 02 - Algebra
Description: Contains a full set of notes on an introduction to the principles of algebra. Useful fundamentals for 1st year science/engineering students covers basics of algebra, forming linear equations, quadratic equations and simultaneous equations.
Description: Contains a full set of notes on an introduction to the principles of algebra. Useful fundamentals for 1st year science/engineering students covers basics of algebra, forming linear equations, quadratic equations and simultaneous equations.