3. Pair of Linear Equations in Two Variables

SEBA Class 10 Maths Chapter 3. 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 and Answers

Question : 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, i.e.,

agian , , and i.e., ]

Question : One equation of a pair of dependent linear equations is ,the second equation can be :

(a)

(b)

(c)

(d)

Solution : (d)

[ We have,

]

Question: If the pair of linear equations and has infinitely many solutions , then the values of and are :

(a) 3 and 5 (b) 4 and 8 (c) 4 and 7 (d) 4 and 5

Solution : (b) 4 and 8

[ We have,

So,

and

]

Question : 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 , , i.e., ]

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

(a) (b) (c) (d)

Solution: (a) .

[ We have ,

So,

]

Question : 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 ]

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

(a) (b) (c) – 11 (d) – 7

Solution: (a)

[ We have ,

So,

]

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

(a) 6 , – 1 (b) 2 , 3 (c) 4 , 1 (d)

Solution: (c) 4 , 1

[ Here , and

We have ,

and

from

From , we get ]

Question : A pair of linear equations ; is said to be inconsistent, if

(a)

(b)

(c)

(d)

Solution: (a)

Question : If the pair of linear equations and has infinitely many solutions , then the value of is :

(a) 3 (b) 4 (c) 5 (d) 6

Solution : (d) 6

[ We have,

]

Question : If the lines and represent a pair of coincident lines, then is :

(a) 4.5 (b) 5.6 (c) 12 (d) 9

Solution : (c) 12

[ We have,

So, and

]

Question : The graph of is a line parallel to the -

(a) – axis

(b) – axis

(d) none of these

Solution: (b) – axis .

Question : 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

So,

i.e., ]

Question : 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 .

Question : 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 , i.e., ]

Question : 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) .

Question : 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.

Question : 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 . ]

Question : Which of the following pair of linear equation has no solution ?

(a) ;

(b) ;

(d) ;

Solution : (a) ;

[ We have ,

]

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 Maths Chapter 3 .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 .