In this chapter, we shall learn more fundamental rules and sequence of operations applicable to simplifying Algebraic Expressions and solving Algebraic Equations. We have seen that
a). All Algebraic Operations are similar to arithmetic Operations and mainly involve Addition, Subtraction, multiplication and Division.
b). Arithmetic deals with numbers while Algebra deals with variables represented by Alphabets with their coefficients – Algebra Expressions
are mathematical phrases represented by numbers and or variables.
c). An Algebraic Equation has two sides – Left hand side (LHS) expression and Right-hand side (RHS), consist of variables and their coefficients. An Algebraic equation can be solved to find the value of the variable to the given equation.
Basic rules of Algebraic operations:
The following basic laws are applicable to Algebraic Operations:
They are:
Commutative law of Addition: The sum of two numbers is the same, regardless of the order in which they are added.
6 + 4 = 10; 4 + 6 = 10.
6 + 4 = 4 + 6 = 10.
So also, a + b = b + a.
a + b + c = b + c + a = c + a + b.
Commutative law of Multiplication: The product of two numbers is the same regardless of the order in which they are
multiplied
3 x 2 = 2 x 3.
Also, a x b = b x a
Associative Law of addition: The sum of three or more numbers is the same regardless the order in which they are grouped
for addition, 5 + 4 + 3 = (5 + 4) + 3 = 5 + (4 + 3) = 4 + (5 + 3).
So also, p + q + r + s = (p + q +r) + s = p + (q + r + s) = (p + q) + (r + s) etc.,
Commutative law of Multiplication: The product of two numbers is the same regardless of the order in which they are
multiplied
3 x 2 = 2 x 3.
Also, a x b = b x a
Associative Law of Multiplication: The product of three or more numbers is the same regardless of the order in which they are grouped for multiplication 4 x 2 x 3 = (4 x 3) x 2 = 4 x (3 x 2).
Also, a x b x c x d = (a x c) x (b x d) = (b x c) x (a x d) etc.,
Basic properties of equality:
If a = b and c = d, then a + c = b + d and ac = bd.
The above property states that the same number can be added to both sides of the
equality and that both sides of the equation can be multiplied by the same number.
Properties of Negatives:
1-(-a) = a
(-a) b = – (ab) = a(-b)
(-a) (-b) = ab
(-1) a = -a
In an algebraic expression, like terms are those terms that contain the same variables raised to the same power. Also, we can add or subtract like terms only while simplifying algebraic expressions and equations.
The above expression can be simplified in many ways, as indicated below but which are not correct method.
(i). 5 a + 2 a . 4 a – a = (5a + 2a) . (4 a – a) = 7 a x 3a = 21 a²; (or)
(ii). 5 a + 2 a . 4 a – a = {(5 a + 2 a) x 4 a} – a = (7 a x 4 a) – a = 28 a² – a; (or)
(iii). 5 a + 2 a . 4 a – a = {5 a + (2a . 4 a) – a}
= (5 a – a) + (2 a . 4 a) = 4 a + 8 a²; and so on.
(iv) 5 a + 2 a . 4 a – a= 5a+ 8x2 -a= 8x2 +4a (correct method)
Here we are not able to ascertain the correct procedure as there is no clear procedure indicated. It is therefore essential that we must have a defined procedure to simplify and solve the problems. This procedure will enable us to arrive at the desired solution and there will be no confusion – such a procedure is called as “ORDER OF OPERATIONS “. This “ORDER OF OPERATIONS “is a standard procedure that specifies the manner and order in which the problem can be simplified or solved and there will be no confusion for anyone.
The “ORDER OF OPERATIONS “is PEMDAS prescribed below:
Step 1: Evaluate and club all the expressions within the Parentheses and other brackets first
Step 2: Simplify and assess all expressions involving Exponents;
Step 3: Do all the remaining Multiplication & Division, as one comes to them, when working from left to right in the expressions;
Step 4: Finally, carry out the remaining Addition & Subtraction, as you approach them, when working from left to right in the expression.
- Simplify the following expression: [{(5 x 3 a) + 9 a} ÷ 12] ²
Step 1: simplify and group all the terms in the parenthesis and brackets first – we get
[{(5 x 3 a) + 9 a} ÷ 12] ² = [{15 a + 9 a} ÷ 12] ²
= [24 a ÷ 12] ²
= [2 a] ²
Step 2 : simplify and expand the exponents. We get
[{(5 x 3 a) + 9 a} ÷ 12] ² = [2 a] ² = 4 a²
2. Simplify: (6 x2y ÷ 2x) + (3x – x)2
Step 1: First simplify the terms in the parentheses – we get,
{(6 x2y ÷ 2x) + (3x – x)2} = (6x²y ÷2x) + (2x) ²
= {(3xy) + (2x) ²}
Step 2: Now remove the parentheses and expand the exponent terms, we get
{(3xy) + (2x) ²} = 3xy + 4 x²
Step 3 : Remove the parenthesis,
we get (6 x2y ÷ 2x) + (3x – x)2 = 3 x y + 4 x² = 3 x y + 4 x ².