Algebra deals with mathematical symbols, arithmetic operations, and variables to represent a situation. It is a branch of mathematics where the equations describe the connection between symbols and variables, just like sentences describe the connection between words.
This article will describe in detail various algebraic identities properties, rules of algebra, formulas and equations.
Algebraic equations are very similar to the properties of numerical equations and expressions. The set of rules governing addition and multiplication apply similarly in algebra.
An equation of algebra mainly consists of four parts: coefficients, variables, exponents and constants.
Take an expression, for example, ax2 + bx + c = d.
Here, ‘a’ and ‘b’ are the coefficients, ‘x’ is the variable, ‘2’ is the exponent, ‘c’ and ‘d’ are the constants. The basic rule of algebra is that the term with the highest exponent is written first, and then the rest of the terms follow with reducing power.
The value of variables is never fixed; it varies depending on the rest of the equation. This example has four terms with like terms (x). An algebraic expression can have many terms, both like and unlike terms.
Here, five basic rules of algebra are described in detail:
Commutative Rule of Addition
In this rule of algebra, the order of variables written in an expression does not matter. The variables can be rearranged anyway, and it still would not affect the answer. For example,
- (a + b) = (b + a)
- (x3 + 3x) = (3x + x3)
Associative Rule of Addition
In this rule, if three or more terms are added in an equation, the order in how they are arranged does not matter. The result is not affected by the reordering of the terms in the algebraic equation. For example,
- (a + b) + c = a + (b + c)
- X5 + (4y + 2 + 3) = (x5 + 4y + 2) + 3
Commutative Rule of Multiplication
The commutative rule of multiplication says that the order in which two algebraic terms are multiplied is changeable. The result is not affected by the reordering of the terms while multiplying. For example,
- (x * y) = (y * x)
- (x4 – 4x) × 5x = 5x × (x4 – 4x),
If we solve this equation, LHS = (5×5 – 20×2)
RHS = (5×5 – 20×2); hence, LHS = RHS, which proves that the order of the algebraic terms does not alter the result.
Associative Rule of Multiplication
In this algebraic rule, the ordering of terms does not matter while multiplying more than three terms. The output remains unaffected by the rearrangement of terms, for example:
- (a * b) * c = a * (b * c)
- x3 × (3×4 × 2x) = (x3 × 3×4) × 2x
Distributive Rule of Multiplication
The distributive rule of multiplication is slightly different from the other four. It says when a term is being multiplied by the addition of two terms, the output is the total sum of the products of two algebraic terms, meaning the term outside of the sum of two terms will be multiplied individually. For example,
- x * (y z) = (x * y) (x * z)
- x2 × (3x y) = (x2 × 3x) (x2× y)
Four basic algebraic operations are no different from numerical operations; the only difference is that we categorise each operation according to the like and unlike terms:
Like numerical addition operations, the algebraic terms are separated by ‘+’. In an algebraic equation, there are either like terms or a mixture of like and unlike terms. The like terms are always added to each other, while the unlike terms are kept separately because two different variables are treated as different quantities. And mathematically, two separate variables which carry different identities cannot be added. For example,
- Like terms: 2a + 5a = 7a, here, ‘a’ is a common variable with different coefficients, making all the terms – like terms.
- Unlike terms: 3a + 5b, here, ‘a’ and ‘b’ are different variables that cannot be added to each other.
These examples state how terms consisting of common variables can be added to each other, while equations with unlike terms do not add further.
Similarly, in the case of subtraction, the algebraic terms are separated by ‘ – ’. This is very similar to the basic numerical subtraction operation where the lower value is subtracted from the higher one. In this case, we have to keep in mind the categorisation of like and unlike terms. Only an equation consisting of like terms can be subtracted. In case, unlike terms, no operations can be performed further. For example,
- Unlike terms: 6xc – 9ay, here, the variables are ‘xc’ and ‘ay’, which are different quantities and cannot be added further.
- Like terms: 5×2 – 3×2 = 2×2
The algebraic multiplication operation states that two or more terms are multiplied when they are separated by ‘×’. Unlike the addition and subtraction operations, it does not matter if we are multiplying like or unlike terms. In this operation, we use the law of exponents. For example,
- Unlike terms multiplication: 6bc * 2×2 = 12bcx2
- Like terms multiplication: 4y * 3y = 12y2
Division algebraic operation is very similar to numerical division operations. If two terms are divided by the ‘/’ sign, the operation performed on them is division. Here, we need to simplify the two terms that are to be divided. If further simplification is not possible on the terms, then they cannot be divided. For example,
- Like terms division: 6x / 3x = 2
- Unlike terms division: 8abc / 2ac = 4b
To perform simple operations on algebraic equations, here are some common formulas that must be remembered for simple algebraic operations:
- (x + y)2 = x2 + 2xy + y2
- (x – y)2 = x2 – 2xy + y2
- (x + y) (x – y) = x2 – y2
- (x +y)3 = x3 + 3x2y + 3xy2 + y3
- (x – y)3 = x3 – 3x2y + 3xy2 – y3
- x3 + y3 = (x + y)(x2 + xy + y2)
- x3 – y3 = (x – y) (x2 + xy + x2)
- (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
While solving any algebraic equation identifying the relationship between two terms is necessary to further explore the expression. If the values of all but one variable are known, they can be substituted in the equation so the value of the unknown variable can be found out. For example, x = 2 in an equation x 2y = 4, after substituting the value we get, 2 2y = 4 ; 2y = 2 ; y = 1. Hence, the value of the unknown variable ‘y’ is calculated.