Algebra Basics
This branch of mathematics, which uses letters or symbols to represent numbers, is very powerful as it allows statements to be made that are true for all cases, not just specific ones.
Vocabulary
constant
This is a value, usually a number, which does not change.
Examples
3 8 ∏
Variable
A variable is a symbol, usually a letter, standing for any value from a given range of numbers.
Examples
x y a z
coefficient
This is a constant in front of a variable.
The variable is multiplied by the constant.
A coefficient of 1 is usually not written.
Examples
3x means
3 times x
27y2 means 27 times y times y
z means
1 times z.
expression
An expression is a list of constants and variables joined together by the operators + - x ÷
Examples
45a - 27b
15x + 9y
(15/c) x 3a
terms
Each part of an expression is called a term
Examples
45x +32x2
has two terms
45x +32x2-9
has three terms
Operations
Collect like terms
Like terms have the same variable.
When we collect like terms, we gather together all the terms that are the same.
example
x+ 5 + 8 + Q + 9 +M + x + Q+ x+Q + Q + Q + x
re arranged becomes
x+x+x+x+ Q+ Q+ Q+ Q+ Q+M+5 + 8 + 9
=4x + 5Q + M + 22
Simplify
All the like terms are collected together by addition or subtraction .
Usually, small letters are used.
The order is coefficients, variables, ordinary numbers
Examples
4x + 17x
= 21x
a + 6b +23a – 3
=24a +6b -3
3a +6b+6-2a +3b+4
= a +9b +10
Evaluate
Substitute the given values into the expression and calculate a result.
Don’t forget the order of operations !
example
if a = 2 and b = 3
a + 2b
= 2 + 2 x 3
= 2 +6
= 8
and
a2 - 3b + 5
= 2x2 – 3x3 +5
= 4 - 9 + 5
= 0
Equation
An equation sets two expressions equal to each other, so that they have the same value. One of the expressions may be a constant.
2b +3 = 7
3(a+b) =3a + 3b
y = x + 2
y = mx + c
Inequation
An inequation occurs when there is an inequality between the expressions.
3x +4 < 7
12x ≥ 60
Satisfy an equation
An equation is satisfied if a value is substituted for a variable and the statement is still true.
The left hand side of the equation must equal the right hand side.
b = 2 satisfies
2b +3 = 7
Since
2x2 + 3
=4+3
=7
b = 3 does not satisfy
2b +3 = 7
Since
2x3 + 3
=6+3
=10
solution
The solution is the value, or set of values, which satisfies the equation
2b +3 = 7
has solution
b = 2
Formulae
A formula is a statement, usually an equation, which allows quantities to be calculated for given values.
Formulae can be expressed in words :-
Example
The cost of tuition is £20 per hour, plus a travel fee of £5.
How much does 6 hours tuition cost ?
Cost of tuition =Number of hours x £20 + travel fee
=6 x £20 + £5
= £120 + £5
= £125
6 hours of tuition costs £125
or in symbols :-
Let £C = cost of tuition and h = number of hours
Then
£C =Number of hours x £20 + travel fee
£C =h x 20 + 5
£C =20h + 5
£C =20 x6 + 5
£C = 120 + 5
£C = 125
6 hours of tuition costs £125
Evaluating Formulae
Example
The formula
Ek = ½mv2
is used to calculate the kinetic energy (Ek) of an object with mass (m) kilograms traveling at speed (v) metres per second.
Use the formula to calculate the
energy of an object of mass 5 kg traveling
at speed 12 m/s .
Ek = ½mv2
= ½x 5 x 12 x 12
= ½x 5 x 12 x 12
= ½x 720
= 360 Joules
Changing the subject of a formula
© Alexander Forrest