3. Pair of Linear Equations in Two Variables

Pair of Linear Equations in Two Variables

Chapter 3. Pair of Linear Equations in Two Variables

Class 10 Maths Chapter 3: Pair of Linear Equations in Two Variables Multiple Choice Questions , Answer following the Questions , Fill in the blanks , 2 Marks Questions , 3 Marks Question , 4 Marks and Solutions :

Class 10 Pair of Linear Equations in Two Variables Multiple choice Questions and Answers

SECTION = A

Q1. Consider the following pairs of linear equations : [SEBA 2020]

(i) ;

(ii) ;

Choose the correct alternative :

(a) The pair in (i) has no solution, whereas the pair in (ii) has unique solution .

(b) The pair in (i) has infinitely many solutions, whereas the pair in (ii) has no solution .

(d) The pair in (i) has no solution, whereas the pair in (ii) has infinitely many solutions .

Solution : (d) The pair in (i) has no solution, whereas the pair in (ii) has infinitely many solutions .

[ We have , and ]

Q2. If the point lies on the lines represented by both the equations and , then the lines is :

(a) intersecting

(b) coincident

(d) None of these

Solution: (a) intersecting .

[ We have , ]

Q3. The value of for which the pair of linear equations and represents parallel lines is :

(a)

(b)

(c)

(d)

Solution: (a) .

[ We have ,

]

Q4. Consider the following pairs of linear equations :[SEBA 2019]

(i) ,

(ii) ,

Choose the correct alternative .

(a) The pairs in (i) and (ii) are consistent .

(b) The pairs in (i) and (ii) are inconsistent .

(d) The pair in (i) is consistent, whereas the pair in (ii) is inconsistent .

Solution: (d) The pair in (i) is consistent, whereas the pair in (ii) is inconsistent .

[ We have , and ]

Q5. If the lines and are coincident , then the value of is :

(a)

(b)

(d) – 7

Solution: (a)

[ We have , ]

Q6. If , is the solution of the equations and , then the values of and are respectively :

(a) 6 , – 1

(b) 2 , 3

(d) ,

Solution: (c) 4 , 1

[ Here , and

We have ,

and

from

From , we get ]

Q7. A pair of linear equations ; is said to be inconsistent, if

(a)

(b)

(c)

(d)

Solution: (a) 2

Q8. The graph of is a line parallel to the -

(a) – axis

(b) – axis

(d) none of these

Solution: (b) – axis .

Q9. The pair of linear equations and is : [CBSE 2020 standard]

(a) consistent

(b) inconsistent

(d) Consistent with many solutions

Solution: (b) inconsistent .

[ We have ,

and

]

Q10. The graph of is a line :

(a) parallel to – axis

(b) perpendicular to – axis

(d) passing through the origin .

Solution: (d) passing through the origin .

Q11. The lines representing the linear equations and are :

(a) intersect at a point

(b) parallel

(d) intersect at exactly two points .

Solution: (b) parallel .

[ We have , ; ]

Q12. The pair of equations and graphically represents lines which are :

(a) Coincident

(b) parallel

(d) intersecting at (4,3)

Solution: (d) intersecting at (4 , 3) .

Q13. If pair of linear equations is consistent , then the lines represented by them are :[CBSE 2020 (Basic)]

(a) always coincident

(b) parallel

(d) intersecting or coincident.

Solution: (d) intersecting or coincident.

Q14. Which of the following pair of linear equations is intersect at a point ?

(a) ,

(b) ,

(d) ,

Solution: (d) , .

[ We have,

So, the pair of linear equations is intersect at a point . ]

Class 10 Pair of Linear Equations in Two Variables Fill in the blanks

Q1. If in the equation , the value of is 6, then the value of will be .

Solution: – 2

[ We have,

]

Q2. If the line are parallel , then the pair of the pair of equation is . [ consistent / inconsistent / dependent (consistent)]

Solution: inconsistent .

Q3. The value of for which equationsand has a no solution is .

Solution: 6

[ We know that,

So,

]

Q4. The solution of the pair of linear equations and are and respectively.

Solution: 2 and – 3

[ We have,

and

{from (i)}

Putting in equation , we get ]

Q5. If and is a solution of a pair of equations and , then the value of and are and respectively.

Solution: 5 and 15

[ Given , and

So,

and

]

Q6. If a pair of linear equations and is dependent and consistent , then the situations can arise .

/ /

Solution: .

Q7. 10 students of class X took part in a Mathematics quiz . If the number of girls is 4 more than the number of boys , then the number of boys and girls who took part in the quiz are and .

Solution: 3 and 7 .

[ let and be number of girls an boys respectively .

A/Q ,

And

From

From , we get

]

Class 10 Pair of Linear Equations in Two Variables Answer following the question :

Q1. Find the value of for which the given pair of linear equations has infinite many solutions : ;

Solution: We have , and

From part and part , we get

Q2. On comparing the ratios , and , find out whether the lines representing the pairs of linear equations intersect at a point , are parallel or coincident :

;

Solution: We have , and

Here , , , , , ,

, and

Thus, the pairs of linear equations are parallel .

Q3. Find the value of so that the point ,lie on the line represented by .

Solution: Here , ,

We have ,

Q4. Find the number of solutions of the following pair of linear equations : [CBSE 2009]

and

Solution: We have , and

Here , , , , , ,

, and

Thus, the pairs of linear equations are infinitely many solutions .

Q5. Write whether the following pair of linear equations is consistent or inconsistent :

and

Solution: We have , ;

Here , , , c1=6 , , ,

Thus, the pairs of linear equations is consistent .

Q6. Solve for and ( Using elimination method) :

;

Solution: We have ,

and

From we get ,

Q7. Which of the following pairs of linear equations has unique solution , no solution , or infinitely many solutions ?

;

Solution: We have, and

Here , , , , , ,

, and

Thus, the pairs of linear equations has infinitely many solutions .

Q8. For what value of does the pair of equations given below has a unique solution ?

;

Solution: We have , and

Here , , , , , ,

Therefore, for all values of , except , the given pair of equations will have a unique solution .

Class 10 Pair of Linear Equations in Two Variables 2 Marks Quesions and Answers :

SECTION = B

Q1. 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 age of Sonu and Nuri respectively .

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

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

A/Q ,

and

[ From ]

From we get ,

Therefore , 20 years and 50 years are the age of Sonu and Nuri respectively .

Q2. Solve : ; andhence find the value of for which .

Solution: We have ,

And

From we get

Q3. In a , . Find the three angles .

Solution: Given,

and

In , we have

, ,

Q4. The difference between two numbers is 26 and one number is three times the other . Find them.

Solution: let and be the two number .

A/Q ,

And

[ From ]

From we get ,

Therefore , the two numbers are 39 and 13 respectively .

Q5. 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 . [SEBA 2019]

Solution: : let and (in metres) are the length and width of the rectangular garden respectively .

A/Q ,

And

[ From ]

From we get ,

Therefore, 20 m and 16 m are the length and width of the rectangular garden respectively .

Q6. For what value of will the following pair of linear equations has infinitely many solutions : ;

Solution: We have , ;

Here , , , , , ,

From 1st part and 2nd part , we get

From 2nd part and 3rd part , we get

Therefore , the value of is 6 .

Q7. If the sum of two positive numbers is 44 and one number is three times the other number, then find the numbers.

Solution : Let and be the first and second numbers respectively .

AQ ,

and

Putting in equation , we get

Therefore, the two positive number are 33 and 11 respectively .

Q8. Solve for x and y : ; [CBSE 2011]

Solution: We have ,

and

Therefore, the value of and are 3 and 2 respectively .

Q9. Solve the following pair of equations by substitution method : ;

Solution: We have ,

and

Putting in equation we get ,

Therefore , the solution are and

Q10. Use elimination method to find all possible solution of the following pair of linear equations : ;

Solution: We have ,

and

, which is a false statement .

Therefore , the pair of linear equations has no solution .

Q11. Given the linear equation , write another linear equation in these two variables such that the geometrical representation of the pair so formed is : (i) intersecting lines (ii) parallel lines

Solution: Given the linear equation is

(i) For intersecting lines ,

Then, the another linear equation is .

(ii) For parallel lines ,

Then , the linear equation is .

Q12. Graphically , find whether the following pair of equations has no solution, unique solution or infinitely many solutions : ;

Solution: We have ,

and

Equation and are same . Hence , the lines represented by equation and are coincident .

Therefore , equation and have infinitely many solutions.

Q13. Solve x and y : ;

Solution: We have ,

and

Therefore, the value of and are 2 and 1 respectively .

Q14. 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 . [SEBA 2020]

Solution: let and be the cost of one pencil and one pen respectively .

A/Q ,

From we get ,

Therefore , the cost of one pencil and one pen are Rs. 3 and Rs. 5 respectively .

Class 10 Pair of Linear Equations in Two Variables 3 marks questions and Answers

SECTION = C

Q1. Solve the pair of equations : [CBSE 2020 standard]

;

Solution: We have,

and

let, and

Putting in equation , we have

and

Therefore, the solutions are : and

Q2. Solve the following pairs of equations by reducing them to a pair of linear equations :

;

Solution: We have ,

;

let , and

and

From we get ,

and

Therefore, the value of and are 4 and 9 respectively .

Q3. Solve for and :

;

Solution: We have,

and

From we get ,

and .

Q4. Solve the following pairs of equations :

; ,

Solution : Let

and

Putting in equation , we get

Hence, and are the required solution of the given pair of equations .

Q5. A fraction becomes when 1 is subtracted from the numerator and it becomes when 8 is a added to its denominator . Find the fraction . [CBSE 2020]

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

So, the fraction is .

A/Q,

and

Putting in equation , we get

Therefore, the fraction is .

Q6. A fraction becomes , if 2 is added to both the numerator and the denominator . If 3 is added to both the numerator and denominator it becomes . Find the fraction . [ SEBA 2016 ,20]

Solution : let, and are the numerator and the denominator of the fraction respectively .

Therefore, the fraction is .

A/Q ,

and

Putting in equation , we get

Required the fraction is .

Q7. Solve for and : [CBSE 2004 , 07C , 08]

Solution: We have ,

and

Putting in equation , we have

Therefore, the solutions are and .

Class 10 Pair of Linear Equations in Two Variables 4 marks Questions and Answers

SECTION = D

Q1. 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 digits of the number respectively.

Therefore, the first number is and when the digits are reversed , then the number is .

A/Q,

And

Putting in equation , we get

Thus , the number .

Q2. Solve the following pairs of equations by reducing them to a pair of linear equations : ; [SEBA 2017 , 19]

Solution: Let and

We have,

and

Putting in equation we get ,

Therefore,

and

Q3. Solve the following pairs of equations by reducing them to a pair of linear equations :

;

Solution : Let and

We have,

and

Putting the value of in equation , we get

Therefore,

and

Q4. 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

Putting in equation we get

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

Class 10 Pair of Linear Equations in Two Variables 5 marks Questions and Answers

SECTION = E

Q1. It can take 12 hours to fill a swimming pool using two pipes . If the pipe of largest diameter is used for 4 hours and the pipe of smallest diameter for 9 hours, only half the pool can be filled. How long would it take for each pipe to fill the pool separately ?

Solution: let and (in hours) be the time taken by the pipe of larger diameter and smallest diameter to fill the pool respectively .

In 1 hour, the pipe of larger diameter fills is .

and in 1 hour, the pipe of smaller diameter fills is .

A/Q, and

let and

and

[ From ]

Putting in , we get

and

So, the pipe of larger diameter alone can fill the pool 20 hours and the pipe of smaller diameter alone can fill the pool in 30 hours .

Q2. Solve for and : ;

Solution: We have ,

and

Let, and

and

[ From ]

Putting in , we get

and

Hence , and is the required solution of the given pair of equations .

Q3. Roohi travels 300 km to her home party by train and partly by bus . He takes 4 hours if she travels 60 km by train and the rest by the bus . If she travels 100 km by train and the remaining by bus ,she takes 10 minutes longer . Find the speed of the train and the bus separately .

Solution: let and are the speed of the train and the bus respectively .

A/Q ,

and

Let, and

and

Putting the value of in , we get

and

Therefore, the speed of the train and the bus are 60 km/hrs and 80 Km/hrs.

Q4. A boat goes 30 km upstream and 44 km downstream in 10 hours . In 13 hours, it can go 40 km upstream and 55 km down-stream . Determine the speed of the stream and that of the boat in still water .

Solution : let, and (in km/h) be the speed of the boat in still water and the speed of the stream .

Therefore, the speed of the boat downstream km/h and the speed of the boat upstream Km/h

A/Q , 10 and 13

let, and

and

Putting the value of in Eq. we get ,

Now,

and

Hence, the speed of the boat in still water is 8 km/h and the speed of the stream is 3 km/h .

Q6. 2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days . Find the time taken by 1 women alone to finish the work, and also that taken by 1 man alone . [ SEBA 2016]

Solution: let time taken by 1 woman and 1 man to finish the work and days respectively .

A/Q ,

and

let and

and

Putting the value of in equation we get

and

Thus time taken by 1 woman and 1 man to finish the work 18 days and 36 days respectively .

Q7. Draw the graphs of the equation and . Determine the coordinate of the vertices of the triangle formed by these lines and the x-axis, and shaded the triangular region.

Solution: We have ,

	– 1	0	1
	0	1	2

and

	4	0	2
	0	6	3

Plot the points A( – 1,0) , B(0,1) , C(1,2) ,D(4,0) , E(0,6) and F(2,3) on graph paper, and join the points to form the lines PQ and RS as shown in figure . We get the shaded triangle AFD with vertices A(– 1, 0) , F(2,3) and D(4,0) .

Q8. Solve the following pair of equation by reducing them to a pair of linear equations:

;

Solution: We have ,

and

Let, and

;

and

Hence, and is the required solution of the given pair of equations .