3. PAIR OF LINEAR EQUATIONS IN TWO VARIABLE (NCERT)

CBSE Class 10 Chapter 3. PAIR OF LINEAR EQUATIONS IN TWO VARIABLE

Chapter 3. Pair of Linear Equations in Two Variables

Chapter 3. Pair of Linear Equations in two variables

Exercise 3.1 complete solution

Exercise 3.2 complete solution

Exercise 3.3 complete solution

Important Note :

1. The general form for a pair of linear equations in two variables x and y is and where, are all real numbers and .

2. Two lines in a plane , only one of the following three possibilities can happen :

(i) The two lines will intersect at one point .

(ii) The two lines will not intersect, i.e., they are parallel .

(iii) The two lines will be coincident .

3. Graphical Method :
The graph of a pair of linear equations in two variables is represented by two lines.
(i) If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is consistent.
(ii) If the lines coincide, then there are infinitely many solutions - each point on the line being a solution. In this case, the pair of equations is dependent (consistent).
(iii) If the lines are parallel, then the pair of equations has no solution. In this case, the pair of equations is inconsistent.

Pair of lines	//Compare the ratios// Graphical representation// Algebraic interpretation//
1. ;	//Intersecting lines // Exactly one solution (unique) // Consistent//
2. ;	// Coincident lines // Infinitely many solutions // Consistent//
3. ;	// Parallel lines // No solution.// Inconsistent//

Class 10 Maths Chapter 3. Pair of Linear Equations in Two Variables Exercise 3.1 Solutions :

1. Form the pair of linear equations in the following problems,and find their solutions graphically.

(i) 10 students of Class X took part in a Mathematics quiz . If the number of girls is 4 more than the number of boys , find the number of boys and girls who took part in the quiz .

Solution : Let and be number of girls and boys respectively .

A/Q ,

	0	10	5
	10	0	5

And

	4	0	5
	0	– 4	1

Photo graph

(ii) 5 pencils and 7 pens together cost Rs 50 , whereas 7 pencils and 5 pens together cost Rs 46 . Find the cost of one pencil and that of one pen .

Solution: Let and be cost of one pencil and one pen respectively .

A/Q ,

	10	3	– 4
	0	5	10

And

	3	8	– 2
	5	– 2	12

Photo graph

2. On comparing the ratios and , find out whether the lines representing the following pairs of linear equations intersect at a point , are parallel or coincident :

(i)

(ii)

(iii)

Solution : (i)

So ,

Therefore, the pair of linear equations are intersect at a point .

(ii)

Here,

So ,

Therefore, the pair of linear equations is coincident .

(iii)

Here,

So,

Therefore, the pair of linear equations are parallel .

3. On comparing the ratios and, find out whether the following pairs of linear equations are consistent or inconsistent .

(i)

Solution : Here,

So,

Therefore , the pairs of linear equations are consistent .

(ii)

Solution : Here,

So,

Therefore , the pairs of linear equations are inconsistent .

(iii)

Solution : We have,

and

Here,

So,

Therefore , the pairs of linear equations are consistent .

(iv)

Solution : Here,

So,

Therefore , the pairs of linear equations are consistent .

(v)

Solution : We have,

and

Here,

So,

Therefore , the pairs of linear equations are consistent .

4. Which of the following pairs of linear equations are consistent/inconsistent ? If consistent, obtain the solution graphically :

(i)

Solution : Here,

So,

Therefore , the pairs of linear equations are consistent .

	5	0	3
	0	5	2

graph

(ii)

Solution : Here,

So,

Therefore , the pairs of linear equations are inconsistent .

(iii)

Solution : Here,

So,

Therefore , the pairs of linear equations are consistent .

	3	0	2
	0	6	2

graph

And

	0	1	2
	– 2	0	2

graph

(iv)

Solution : Here,

So,

Therefore , the pairs of linear equations are inconsistent .

5. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m . Find the dimensions of the garden .

Solution : Let and be the length and width of the rectangular garden respectively .

A/Q ,

	20	16	21
	16	20	15

And

	0	– 4	2
	– 4	0	– 2

Graph

6. Given the linear equation , write another linear equation in two variables such that the geometrical representation of the pair so formed is :

(i) intersecting lines (ii) parallel lines (iii) coincident lines

Solution : (i) intersecting lines :

Thus , the another linear equation in two variable is

[ ]

(ii) parallel lines :

Thus , the another linear equation in two variable is

[ ]

(iii) coincident lines :

Thus , the another linear equation in two variable is

[ ]

7. Draw the graphs of the equations and . Determine the coordinates of the vertices of the triangle formed by these lines and the -axis , and shade the triangular region .

Solution : We have ,

	0	– 4	2
	– 4	0	– 2

and

	4	0	2
	0	6	3

Graph

Class 10 Maths Chapter 3. Pair of Linear Equations in Two Variables Exercise 3.2 Solutions :

1. Solve the following pair of linear equations by the substitution method .

(i)

Solution : We have ,

And

[ From ]

Putting the value of in eq. , we get

Therefore, the solution is and .

(ii)

Solution : We have ,

And

[ From ]

Putting the value of in equation , we get

Therefore, the solution is and

(iii)

Solution : We have ,

And

[ From ]

Which is a false statement . Therefore , the equation do not have a common solution . So, the two rails will not cross each other .

(iv)

Solution : We have ,

And

[ From ]

Putting the value of in equation , we get

Therefore, the solution is and .

(v)

Solution : We have,

And

[ From ]

Putting the value of in equation , we have

Therefore, the solution is and .

(vi)

Solution : We have,

and

Putting the value of in equation , we have

Therefore, the solution is and

2. Solve and and hence find the value of for which .

Solution : We have ,

and

Putting in equation , we get

Therefore, the value of m is – 1 .

3. From the pair of linear equations for the following problems and find their solution by substitution method .

(i) The difference between two numbers is 26 and one number is three times the other . Find them .

Solution : Let and be the two numbers .

A/Q ,

And

Putting the value of in equation , we have

Therefore , the two numbers are 39 and 13 .

(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees . Find them .

Solution : Let and be the two angles .

A/Q ,

And

[ From ]

Putting the value of in equation , we get

Therefore, the angles are and respectively .

(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs 3800 . Later, she buys 3 bats and 5 balls for Rs 1750 . Find the cost of each bat and each ball .

Solution : Let and be the cost of one bat and one ball respectively .

A/Q ,

and

[ From ]

Putting the value ofin equation , we get

Therefore , the cost of one bat and one ball are Rs 500 and Rs 50 respectively .

(iv) The taxi charges in a city consist of a fixed charge together with the charge for the distance covered . For a distance of 10 km, the charge paid is Rs 105 and for a journey of 15 km, the charge paid is Rs 155 . What are the fixed charges and the charge per km ? How much does a person have to pay for travelling a distance of 25 km ?

Solution : Let, and be the fixed charge and the charge per km respectively .

A/Q ,

And

[ From ]

Putting in equation we get

Therefore, a person have to pay for travelling a distance of 25 km Rs. ( ) Rs.( ) Rs.( ) Rs.

(v) 29. A fraction becomes , if 2 is added to both the numerator and the denominator . If 3 is added to both the numerator and the denominator it becomes . Find the fraction .

Solution : Let, and are the numerator and the denominator of the fraction and the fraction is .

A/Q ,

and

Putting in equation , we get

Therefore , the fraction is .

(vi) Five years hence, the age of Jacob will be three times that of his son . Five years ago , Jocob’s age was seven times that of his son . What are their present ages ?

Solution : Let and be the present age of the Jacob and his son respectively .

Five years hence , the age of Jacob and his son will be and years respectively .

Five years ago , the age of Jacob and his son will be and years respectively .

A/Q ,

And

[ From ]

Putting the value of in equation , we get

Therefore , 40 yrs and 10 yrs are the present age of the Jacob and his son respectively .

Class 10 Maths Chapter 3. Pair of Linear Equations in Two Variables Exercise 3.3 Solutions :

1. Solve the following pair of linear equations by the elimination method and the substitution method :

(i)

Solution : Using elimination method :

And

Putting the value of in equation , we get

Therefore , the solution is and

Using substitution method :

And

Putting the value of in equation , we get

Therefore , the solution is and

(ii)

Solution : Using elimination method :

And

Putting the value of in equation , we have

Therefore, the solution is and .

Using substitution method :

And

Putting the value of in equation , we get

Therefore, the solution is and .

(iii)

Solution : Using elimination method :

We have ,

And

Putting the value of in equation , we get

Therefore, the solution is and

Using substitution method : We have ,

And

Putting the value of in equation we get

Therefore, the solution is and .

(iv)

Solution : Using elimination method :

We have,

and

Putting in equation , we have

Therefore, the solution is and

Using substitution method : We have

We have,

and

Putting the value of in equation , we get

Therefore, the solution is and

2. Form the pair of linear equations in the following problem , and find their solutions (if they exist) by the elimination method :

(i) If we add 1 to the numerator and subtract 1 from the denominator , a fraction reduces to 1 . It becomes if we only add 1 to the denominator . What is the fraction ?

Solution : Let and be the numerator and denominator of the fraction .

The fraction is .

A/Q ,

and

Putting the value of in equation , we get

Therefore , the fraction is .

(ii) Five years ago , Nuri was thrice as old as Sonu . Ten years later, Nuri will be twice as old as Sonu . How old are Nuri and Sonu ?

Solution : Let and be the present age of the Nuri and Sonu respectively .

Five years ago , the age of Nuri and Sonu will be and years respectively .

Five years later , the age of Nuri and Sonu will be and years respectively .

A/Q ,

And

Putting the value of in equation , we get

Therefore , 35 yrs and 15 yrs be the present age of the Nuri and Sonu respectively .

(iii) The sum of the digits of a two-digit number is 9 . Also , nine times this number is twice the number obtained by reversing the order of the digits . Find the number .

Solution : Let and be the ten’s and the unit’s digit of the number respectively .

The number is .

When the digits are reversed , then the

number is .

A/Q ,

Putting the value of in equation , we get

Therefore , the number is 18 .

[ ]

(iv) Meena went to a bank to withdraw Rs 2000 . She asked the cashier to give her Rs 50 and Rs 100 notes only . Meena got 25 notes in all . Find how many notes of Rs 50 and Rs 100 she received .

Solution : Let and be the number of notes of Rs 50 and Rs 100 respectively .

A/Q ,

Putting the value of in equation , we get

Therefore , 10 and 15 be the number of notes of Rs 50 and Rs 100 respectively .

(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter . Saritha paid Rs 27 for a book kept for seven days , while Susy paid Rs 21 for the book she kept for five days . Find the fixed charge and the charge for each extra day .

Solution : let, and be the fixed charge and the charges for each extra day .

A/Q ,

And

Putting the value of in equation , we get

Therefore, the fixed charge is Rs 13 and the charge for each extra day is Rs 3 .