All about Linear Equation in Two Variables and pair of Linear Equation in Two Variables

 

Linear Equation in Two Variables 

pair of Linear Equation in Two Variables

 

What is Linear Equation in Two Variables?

Linear equation is a mathematical equation which can be represented in the form of the equation ax+ by+ c = 0 is called linear equation in two variables.

Where a, b, c are real number.  Both a and b are not zero at a time.

 

Example:

"Sum of two numbers is 100" we can represent the statement in the form of mathematical equation as x + y = 100 (so, there two variables are x and y). There are infinitely many values of x and correspondingly of y.

 

"Difference of 3 times of A's tuition fee and 2 times of B's tuition fee is Rs.50. For this statement the mathematical equation can be 3A-2B=50

 

Types of Question with Answer:

1.     Express the linear equation 7 = 2x in the form of ax + by + c = 0 and also write the values of a, b, and c.

2.     The cost of a toy telephone is the same as the cost of 3 balls. Express the statement in the form of linear equation in two variables

 

Solution of Linear Equation:

The values of the two variables that satisfy the equation are the solutions of the given linear equation.

 

X+2Y = 10 (If we put x=4 and y=3)

LHS 4 + 2 x 3 = 4 + 6 = 10= RHS

 

(If we put x= 2 and y = 4)

LHS 2 + 2 x 4= 2 + 8 = 10 = 10 RHS

Therefore a linear equation in two variables can have infinitely many solutions.

 

Types of Questions with Answer

 

Find the value of k so that x=1, y= -1 is a solution of 2x +KY = 19. Find two more solutions of resulting equation.

 

Graph of a linear Equation in two variables

2x + 5y = 12, Three solutions for this equation may be (1, 2), (-4, 4) and (6, 0).We can plot these points as A, B and C on graph paper.

 

Pair of Linear Equation in Two Variables

 

We may present a pair of linear equations in two variables as follow a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0.

When we draw the graphs of these two linear equations on the same graph paper, we can have two lines which are

1.     Intersecting at a point.

2.     Coincident to each other.

3.     Parallel to each other.

 

Graphical Method of solution of a pair of linear equations

 

7x-y=42----1

 

x	6	5
y	0	-7

 

3x-y=-6----2

 

x	0	-2
y	6	0

 

Outcome:

1.     When two lines are intersecting the solution will be unique. It means there is only one common solution of both the equations.

2.     If two lines are coinciding there are infinitely many solutions of the pair of equations.

3.     When two lines are parallel to each other then there is no solution of the pair of equations.

 

Therefore when the two lines are either intersecting or coincident we say that the given linear equations are consistent. If both are parallel we say them inconsistent.

Coefficient comparison method to know consistent or inconsistent of two linear equations

 

a1/a2≠b1/b2          Unique Solution   Intersecting lines   Consistent

a1/a2=b1/b2=c1/c2     Infinitely many Solution Coinciding lines Consistent & Dependent

a1/a2=b1/b2≠c1/c2         No solution Parallel lines          Inconsistent

 

Example :

3x+2y=5, 2x-3y=7

a1=3, b1=2, c1=-5

a2=2, b2=-3, c2=-7

a1/a2=3/2, b1/b2=2/(−3), c1/c2=(−5)/(−7)

 

a1/a2≠b1/b2   because 3/2≠2/(−3)

 so the linear pair is consistent.

 

Algebraic method of solving a pair of linear equations

 

Substitution Method

 

In this method we take any one of the given pair of linear equations and express one variable in terms of the other variable,

 

Example:

2x+3y=7  ---(1)

3x+5y=18 ---(2)

 

From (1) y=(7−2x)/3

 

Put the value of Y in (2)

3x+5 ((7−2x)/3)=18

⟹3 x 3x+35 −10x=18 x 3

⟹9x+35 −10x=54

⟹−x=54−35=19

X=-19

Put the value of x in (1)

Y=(7−2 x (−19))/3=(7+38)/3=45/3=15

 

The solution is x= -19, y= 15

 

 

Put the value of x in (1)

Y = (8−2)/3=6/3=2

∴x=1, y=2

 

 

Detail Study of Compound Sentence/ How to form compound sentence of English language. 

 

Definition:

A sentence which is made of many finite clauses or independent clauses is known as compound sentence. A compound sentence is formed combining multiple independent clauses with the help of coordinating conjunctions.

 

Coordinating conjunctions:

And, But, Or, Yet, Still, Nevertheless, Or else, Otherwise, Therefore, Either...Or, Neither...Nor, not only...But also, either...or, both...And, So, for and besides etc. are mostly used conjunctions.

 

Illustration:

He joined the English essay competition and won the 1^st prize.

 

In this above sentence he joined the English essay competition is an independent clause and he won the 1^st prize is another independent clause. The conjunction is and.  The first clause is main clause while the second clause is coordinate clause in this sentence. 

 

More illustrations:

And: He came and stayed for a week.

But: He is really intelligent but he is not hard working.

Yet: He is not rich, yet he is happy.

Still: She did not do hard work still she passed the examination.

Or: He watches television or goes to playground.

Or else: Work hard or else you will fail.

Otherwise: Take care, otherwise, you will fail.

Therefore: He was taken ill; therefore he could not attend the meeting.

Either...or: Either you leave or I will.

Neither...nor: He is neither handsome nor smart.

Not only...but also: He not only liked the home but also bought it.

Both... and: He can both sing and act.

So: I was very hungry, so I wanted to have lunch.

For: He found it increasingly difficult to read, for his eyes were failing.

Besides: I don’t want to go to the cinema, besides I am feeling tired.

 

Combine each pair of the following sentences into a compound sentence

 

1.     He finished his exercises. He put away his books

2.     He must not be late. He will be punished.

3.     Make haste. You will be late.

4.     He is rich. He is not contented.

5.     He was horrified. He saw bloodstain on the floor.

6.     He tried hard. He did not succeed.

7.     They took every precaution. They suffered from the disease.

8.     I have a lot of work. I must do it now.

9.     Be good. You will be happy.

10.                        He practiced daily. He became an expert player

11.                        Energy is necessary for success in life. Patience is also necessary for success in life

12.                        I have left my glasses on the bus. I have left my glasses at the office

13.                        He cannot speak English. He cannot speak Hindi.

14.                        The boy is very healthy. He is also very happy.

15.                        He does not drink tea. He does not drink coffee.

16.                        She draws pictures neatly. She also paints them beautifully.

17.                     The artists will sing for us. They will dance for us too

All about Simple Sentence of English language.// Types and Structure of Simple Sentence of English language.

 

Simple Sentence/ Structure and Types of Simple Sentence

Watch Video about Simple Sentence

 Definition:

A sentence is a group of words taken from parts of speech and arranged grammatically to make a complete sense is called sentence. According to English grammar to form a sentence it needs a subject and a predicate. Broadly, a sentence is formed by combining clauses, phrases, adverbials, words and morphemes, which must produce certain meaning.

 

Example:

The children watched a TV programme while their mother made the tea.

Detail Analysis

The children watched a TV programme while their mother made the tea. (One Sentence).

 “The children watched a TV programme” and “Their mother made the tea” (Two Clauses).

 ‘The children’ and ‘watched a TV programme’ are phrases of first clause. (Phrases).

‘Children’ is a word and ‘child’ and ‘ren’ separately called morphemes. (Word and morphemes).

 

Types of sentence

According to number of clauses used in formation of sentence, the sentences can be of four types. Those are simple, compound, complex and mixed types. In this blog we are going to learn all about simple sentence.

 

Simple sentence:

If a sentence consists of one clause, it is described as simple sentence. It is made of one principal clause only. 

 

Example: 

I play football.

She goes to market.

They make a noise.

We swim in the river.

 

Types of simple sentence:

There are six types of simple sentence. Those are given below.

Assertive sentence: These are positive statement types of sentence. Assertive is also known as declarative and affirmative. Declarative sentence simply states facts and opinions. A full stop (.) is placed at the end of sentence.

Structure:

Subject-Verb group-object/ complement/ adverbials.

Example:

The sun rises in the east. (Fact)

It may rain today. (Opinion)

 

Negative sentence: These are negative statements or sentence of negation type opinion or fact. A full stop (.) is placed at the end of sentence.

 Structure:

Subject-Helping verb- not- Main verb-object/complement/ adverbials.

 Example:

The moon does not appear in the sky every night.

He may not pass in this examination.

 

 Yes/No Question: The sentence used to interrogate (to ask question) is known as interrogative or question. If a question can be answered by saying yes or no only, it comes under yes/no question group. A question mark (?) is placed at the end of sentence.

 Structure:

Helping verb-Subject- Main verb-object/complement/adverbials-?

 Example:

Are you a doctor?    Ans: yes or No

Have you played cricket?  Ans: Yes/ No

 

WH Question: If a question cannot be answered by yes/no and needs additional information to answer is known as WH Question. Such questions begin with WH words like what, where, when, who, which, why, how, how many, how much etc. A question mark (?) is placed at the end of sentence.

Structure:

WH words-Helping verb-Subject- Main verb-object/complement/adverbials-? (Direct question)

 WH words- Subject-Verb group-object/ complement/ adverbials-? (Indirect question)

  

Example:

What is your name?

Where is your classmate?

How do you play chess?

 

Imperative: The sentence used to place order, command, suggestion, advice and request comes in imperative group. Such sentence normally begins with ‘verb one’ and ‘let’. A full stop (.) is placed at the end of sentence.

 Structure:

Verb one-object/complement/ adverbials.

Let-accusative pronoun-verb one- object/complement/ adverbials.

 

 Example:

Come here.

Go to office.

Let me ask a question.

Let them speak loudly.

 Exclamatory: An exclamatory sentence expresses feeling of surprise, disgust, excitement and pain. Such sentence begins with How and What. An exclamation mark (!) is placed at the end of sentence.

 Structure:

How-adverb/adjective-rest of the sentence-!

What-noun phrase- rest of the sentence-!

 Example:

What a beautiful flower the rose is!

How nicely she sings!

How heart-breaking the news is!

 Terms related to sentence.

Subject: The doer or agent of the verb. To find out subject ask question using who? Or what? It can be of one word or multiple words. The head word of subject part may be a noun or pronoun.

 

Predicate: Apart from subject the remaining part of sentence is known as predicate which comprises verb group, objects and complement.

 

Verb and verb group: Verbs can be auxiliary and lexical types. Be, Have, Do and Modals are grouped under auxiliary or helping verb while names of all actions are known as lexical or full verbs. Verb group is formed using helping verb and main verb together.

 

Object: The noun or accusative pronoun in predicate which relate to verb is called object. To find out object put on question “what?” Or “for whom?”  In predicate part of sentence.

 

Complement: It is a type of word that adds more information or describe subject or object in a sentence.

 

Parts of speech: There are eight certain types of words used in English language such as Noun, Pronoun, Adjective, Verb, Adverb, Preposition, Conjunction and Interjection called parts of speech.

 

Clause: A clause contains a subject and a finite verb. A finite verb is one that carries time and changes its appearance.

 

Phrase: A phrase is a part of clause. It does not make complete sense.

 

Adverbials: There are some words or phrases that answer questions such as where, when, how and why are called adverbials.

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