A Short Discussion on Algebraic Equations

Algebraic Expression

A mathematical expression is made up of variables, numbers, and an arithmetic operation between them. For instance, (4m+ 5) is an expression in which terms 4 and 5 are constants and term m is a variable separated by the arithmetic operation +. (plus). There is no fixed value for a variable. Expression variables are typically represented by letters such as a, b, c, m, n, p, x, y, z, and so on. By mixing different variables and numbers, we can create a variety of expressions. In mathematics, an algebraic expression is one that contains variables and constants as well as algebraic operations like addition and subtraction.

Example : 3x + 4y = 7

Types of algebraic expression

Monomial Expression: A monomial is classified as an algebraic expression that has only one term. Ex-  7pq is a two variable monomial in p and q

Binomial Expression : A binomial is an algebraic expression that has two improbable terms. Ex- m+n is two variable binomial in m and n

Trinomial Expression : An algebraic expression of only three non – zero terms is called a trinomial. Ex- x+y+z is a three variable trinomial in x, y and z. 

Polynomial Expression: In general a word with a variable non – negative integral exponent is defined as a polynomial. Ex-2a+5b is a two term polynomial in two variables a and b. 

Algebraic Identities

Algebraic identities are equations that hold true for all possible values of variables. Numbers, variables (unknown values), and mathematical operators make up algebraic identities and expressions (addition, subtraction, multiplication, division, etc

Algebraic identities are utilised in a variety of mathematical fields, including algebra, geometry, and trigonometry. These are primarily used to find the polynomial factors. A deeper understanding of algebraic identities helps to improve the efficiency with which problem sums are solved. The factorisation of polynomials is one of the most important applications of algebraic identities.

What are Algebraic Identities, and how do you use them?

An identity is a mathematical formula that holds true for all possible values of the variables. For all values of the variable, the algebraic identities are equations in which the left-hand side of the equation equals the right-hand side of the equation.

Take the equation ax+b=0 as an example.

When x=–ba, both the left and right sides of the preceding equations are equal. As a result, it’s an equation, not an identity.

We know that (a+b)2=a2+b2+2ab holds for all a and b values. As a result, it’s a persona.

Standard Identities in Algebra

In several disciplines of mathematics, we have some standard identities to employ. The Binomial theorem is used to generate all of the standard identities

In several disciplines of mathematics, we have some standard identities to employ. The Binomial theorem is used to generate all of the standard identities.

The following are four standard algebraic identities. 

Identity-1: Algebraic Identity of Square of Sum of Two Terms (a+b)2 :

 (a+b)2 =a2+2ab+b2

Identity-2 :Algebraic Identity of Square of Difference of Two Terms (a–b)2 :

(a–b)2=a2–2ab+b2

Identity-3: Difference of Two Squares Algebraic Identity

 (a+b)(a–b)=a2–b2

Algebraic Identity (x+a)(x+b)=x2+(a+b)x+ab (x+a)(x+b)=x2+(a+b)x+ab

Algebraic Trinomial Identities

Trinomial algebraic identities are the equivalent equalities. These identities can be derived simply by factoring and manipulating the terms listed below:

1. Identity -1: (a+b)(a+c)(b+c)=(a+b+c)(ab+ac+bc)–abc=(a+b+c)(ab+ac+bc)–ab

2.Identity-2:a3+b3+c3–3abc=(a+b+c)(a2+b2+c2–ab–bc–ca)

3. Identity-3:  a2+b2+c2=(a+b+c)2–2(ab+bc+ca)

The substitution method is used to verify the algebraic identities. Substitute values for the variables and conduct the arithmetic operation with this method. The activity technique is another way to check the algebraic identity. You’ll need a basic understanding of geometry for this procedure, as well as certain materials to prove your identity.

Algebraic identity is the equality that holds regardless of the variable’s value. Variables, numbers, and mathematical operators make up an algebraic expression. When the values of variables are altered, the value of an algebraic expression changes. 

Solve the equation. 

An algebraic equation is one that looks like this:

P = 0

P denotes a polynomial.

For example, x + 8 = 0, where x + 8 is a polynomial, is an algebraic equation. As a result, it’s also known as a polynomial equation.

A balanced equation with variables, coefficients, and constants is always an algebraic equation.

Consider the following equation: 1+1 Equals 2.

It is balanced because both sides have equal weight. Make sure that any modification on one side of the equation is reciprocated on the other side to avoid making an error that throws the equation out of balance. If you want to add a number 5 to one side of an equation, you must also add a number 5 to the opposite side.

1. Simplify the following equation: 2(x+4)+3(x–5)–2y=0

Solution:

2(x+4)+3(x5)–2y=0 given equation

2x+2×4+3x–3×5–2y=0 (Parenthesis are removed using the Distributive 

property.)

2x+8+3x–15–2y=0 (Simplifying)

5x–2y–7=0 (to simplify things even more)

Q.2: Solve x + 1 = 9

x+1 = 9 is the solution.

If we move 1 from the LHS to the RHS, the 1 sign will change.

x = 9 – 1

x = 8

As a result, the answer is.

Question 3. Solve 15 + 5x = 0.

15 + 5x = 0 is the solution.

Here, we must solve for x.

Allow the term with x to remain on the LHS while the rest of the words are moved to the RHS.

5x = 0 – 15

5x = -15

Now multiply both sides by 5.

(5x)/5 = -15/5

x = -3

As a result, the needed solution is x = -3.

Solve -10x – 19 = 19 – 8x in Q.4

The algebraic equation is given as -10x – 19 = 19 – 8x.

The given equation for x must be solved.

On one side, write the terms with x, and on the other side, write the other terms. As a result, we’ll multiply both sides by 8x.

-10 x -19 + 8x = 19 – 8x + 8x

Now group the terms that are similar.

-10x + 8x – 19 = 19

-2x – 19 = 19

Add 19 to both sides now.

-2x – 19 + 19 = 19 + 19

-2x = 38

2 divided on both sides

(-2x)/2 = 38/2

-x = 19

-1 multiply both sides

-x (-1) = 19 (-1)

x = -19

As a result, the needed solution is x = -19.

Conclusion 

In this article we have learnt about algebraic expression. It is defined as the combination of terms by the operations such as addition , subtraction, multiplication ,  division etc. There are various types of algebraic expressions such as monomial expression, binomial expression, trinomial expression and polynomial expression. Algebraic expression can also be divided as numerical expression and variable expression. 

Algebraic identity is an important set of formulas in maths . These are equations where the value of the left  hand side of the equation is identically equal to the value of the right hand side of the equation. 

And we have seen how to solve the equations of algebra.