NCERT Class 10 Chapter 6 Pair of Linear Equations in Two Variables Official CBSE Board Sample Problems Long Answer (For CBSE, ICSE, IAS, NET, NRA 2022)
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Question
4 chairs and 3 tables cost? 2100 and 5 chairs and 2 tables cost? 1750. Find the cost of one chair and one table separately.
Solution
Let the cost of one chair and one table is ′ x and y respectively.
According to question,
Multiplying equation (i) by 2 and equation (ii) by 3, we get
Subtract ing equation (iii) from (iv) , we get
Putting in equation (i) , we get
Hence, cost of one chair = 150 and cost of one table = 500.
Question
Solve for x and y:
and [where
Solution
Given equations are
… . (i)
… . (ii)
Let and then above system becomes.
… (iii)
… . (iv)
On multiplying equation (Eli) by 3 and then adding with equation (ii ′ ) , we get
Putting in equation (iii) , we get
Thus, and and
and
and
Hence, solution of system is and .
Question
Solve for x and y:
Solution
Given equations arc
… (i)
… (ii)
Multiplying equation (1) by ‘a’ and equation (ii) by ‘b’ and then adding, we get
… (i)
… (ii)
Putting in equation (I) , we get
Hence, solution of the system is and
Question
Solve the following pair of equations by reducing them to a pair of linear equations:
Solution
Given system is
… (i)
… (ii)
Iet and then the system becomes,
… (iii)
… (iv)
On multiplying equation (iii) by 3 and equation (iv) by 4 and then adding, we get
Putting in equation (iii) , we get
Thus and
Hence, solution of the system is .
Question
Determine graphically whether the following pair of linear equations has
(i) A unique solution
(ii) Infinitely many solutions or
(iii) No solution
Solution
X | |||
y |
X | |||
y |
Since the lines whose equations are given above intersect at one Point (1, -1) so, given pair of linear equations have a unique solution.
Question
In a two digit number, the digit in the unit place is twice of the digit in the tenth place. If the digits are reversed, the new number is 27 more than the given number. Find the number.
Solution
Let unit՚s place digit be ‘x’ and ten՚s place digit be ‘y’
Then the two digit number
According to 1st condition, … (i)
On reversing the digits . f two digit number, the number becomes .
According to 2nd condition,
… (ii)
Substituting the value of x from equation (i) in equation (IA) , we get
Putting in equation (I) , we get
Hence, the number is 36.
Question
Solve the following pair of linear equations in two variables:
Solution
On solving we get y = b2 and x = a2
Question
A boat covers 32 km upstream and 36 km downstream in 7 hours. In 9 hours it can cover 40 km upstream and 48 km downstream. Find the speed of the stream and that of the boat in still water.
Solution
Let the speed of boat in still water be x km/hr and of the stream be y km/hr.
Let
Solving we get:
Hence
Speed of boat in still water = 10 km/hr
Speed of stream = 2 km/hr
Question
A man travels 600 km partly by train and partly by car. It takes 8 hours and 40 minutes if he travels 320 km by train and the rest by car. It would take 3 minutes more if he travels 200 km by train and the rest by car. Find the speed of the train and the car.
Solution
Let the speed of the train be x km/h and speed of the car be y km/h
These two equations can be simplified as:
On solving these two equations we get km/hr km/hr Speed of train = 80km/hr and speed of car = 60 km/hr
Question
Solve the following system of equations graphically:
Also find the area of the triangle formed between the lines and the x- axis.
Solution
Are square units.
Question
Two places ‘A’ and ‘B’ are 120 km apart from each other on a highway. A car starts from ‘A’ and another from ‘B’ at the same time. If they move in the same direction, they meet in 6 hours and if they move in opposite direction, they meet in 1 hour and 12 minutes. Find the speed of each car.
Solution
Let the speed of the cars be x km/hr and y km/hr respectively where x > y.
Solving we get and
So the speeds of the two cars are 60km/hr and 40km/hr
Question
8 men and 12 boys can finish a piece of work in 5 days, while 6 men and 8 boys can finish it in 7 days. Find the time taken by 1 man alone and that by 1 boy alone to finish the work.
Solution
Let the time taken by a man be x days time taken by a boy be y day՚s work done by 1 man in one day work done by 1 boy in one day
Let
and
and
One man alone can finish the work in 70 days and one boy alone can finish the work in 140 days.
Question
The sum of a two-digit number and the number obtained by reversing the order of its digits is 121, and the two digits differ by 3. Find the number.
Solution
Let no at ones = x
Let no. at tens
New no
Let no. at ones place
Let no. ta tens place 10x
The original no. is 74
The new no. is 47
Question
Solve the following pair of equations:
= 5 and
= 15 where .
Solution
Given simultaneous equations:
… (1)
and
… (2)
Let
Then,
The equations becomes, (3)
… (4)
Now,
Multiply equation (3) by 7, and
Equation (4) by 2, we get
… (5)
(6)
Add equations (5) and (6) , we
Get
Substitute in equation (3) ,
We get
Therefore,
and