Class 10 Exam  >  Class 10 Notes  >  Mathematics (Maths) Class 10  >  Long Answer Questions: Pair of Linear Equations in Two Variables

Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

Q1: For what value of p, the pair of linear equations px = 2y; 2x - y + 5 = 0 has unique solution?
Sol: We have: px = 2y
⇒ p - 2y = 0
2x = y + 5
⇒ 2x - y = - 5
Here, a1 = p, b1 = - 2,
c1 = 0
a= 2, b2 = - 1,
c2 = - 5
For a unique solution,

a1a2b1b2

Substitute the values:

p2-2-1

p2 ≠ 2

p ≠ 4 


Q2: In a cyclic quadrilateral PQRS, ∠P = (2x + 4)°, ∠Q = (y + 3)°, ∠R = (2y + 10)° and ∠S = (4x - 5)°. Find its four angles. 
Sol:
In a cyclic quadrilateral, the sum of opposite angles is 180°.

From ∠P + ∠R = 180°:

(2x + 4) + (2y + 10) = 180 → 2x + 2y = 166  (Equation 1)

From ∠Q + ∠S = 180°:

(y + 3) + (4x - 5) = 180 → 4x + y = 182  (Equation 2)

Solving the equations:

From Equation 1: y = 83 - x

Substitute into Equation 2:

4x + (83 - x) = 182 → 3x + 83 = 182 → x = 33

Find y:

y = 83 - 33 = 50

Calculate the angles:

  • ∠P = 2x + 4 = 70°
  • ∠Q = y + 3 = 53°
  • ∠R = 2y + 10 = 110°
  • ∠S = 4x - 5 = 127°

Therefore,  ∠P = 70°, ∠Q = 53°, ∠R = 110°, ∠S = 127°

Q3: Solve:
23x + 35y = 209
35x + 23y = 197
Sol: We have:
23x + 35y = 209   ...(1)
35x + 23y = 197    ...(2)
58x + 58y = 406 [Adding (1) and (2)]
⇒ x + y = 7 ...(3) [Dividing by whole equation by 58]
Subtracting (1) from (2),
We get, 
12x - 12y = - 12
⇒ x - y = 1 ...(4) [Dividing whole equation  by 12]
Adding (3) and (4),We get, 

x + y + x - y = 7 + 1

⇒ 2x = 8 ⇒ x = 4
Putting the value of x = 4 in (3) 

We get,

x + y = 7 ⇒ 4 + y = 7
⇒ y = 3
So, x = 4 and y = 3.


Q4: Solve:
3x + 5y = 70   ...(1)
7x - 3y = 60   ...(2)
Sol: To solve the equations:

3x + 5y = 70

7x - 3y = 60

Step 1: Eliminate one variable

Multiply Equation (1) by 3 and Equation (2) by 5:

(9x + 15y = 210)

(35x - 15y = 300)

Add the two equations:

(44x = 510)

x = 51044 = 25522

Step 2: Substitute x into Equation (1)

Substitute x = 25522 into 3x + 5y = 70:

3 x 25522 + 5y = 70

76522 + 5y = 70

5y = 70 - 76522

5y = 154022 - 76522

5y = 77522 

y = 775110 = 15522 

Therefore,  x = 25522, y = 15522

Q5: Without drawing the graphs, state whether the following pair of linear equations will represent intersecting lines, coinciding lines or parallel lines:
6x - 3y + 10 = 0
2x - y + 9 = 0
Sol: Here, the given set of equations is:
6x - 3y + 10 = 0   ...(1)
2x - y + 9 = 0    ...(2)
From (1) and (2), we have:
a1 = 6, b1 = - 3, c1 = 10
a2 = 2, b2 = - 1, c2 = 9

Now , a1a2 = 62 = 3

∴ We have: a1a2 = b1b2c1c2

This condition represents parallel lines. Hence, the given pair represents parallel lines.

Q6: Check graphically whether the pair of equations 

3x - 2y + 2 = 0

32 x - y + 3 = 0 

is consistent. Also find the co-ordinates of the points where the graphs of the equations meet the y-axis.

Sol: 3x - 2y + 2 = 0 ⇒ y = Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables ..(1)
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Also = 0 ⇒ y = Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables ...(2)
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Plotting the points (0, 1), (2, 4), (- 2, - 2) and (0, 3), (2, 6), (–2, 0) we get two straight lines l1 and l2 which are parallel.
∴ The given equations are inconsistent.
From the graph, we observe that line l1 meets the y-axis at (0, 1) and line l2 meets the y-axis at (0, 3).
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

Q7: A fraction becomes 1/3 if 2 is added to both numerator and denominator. If 3 is added to both numerator and denominator, it becomes 2/5. Find the fraction.

Sol:  Let the fraction be: xy

From the first condition: x + 2y + 2 = 13

Cross-multiplying:

3(x + 2) = y + 2

Simplify:

3x - y = -4   (1)

From the second condition: x + 3y + 3 = 25

Cross-multiplying:

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

Simplify:

5x - 2y = -9   (2)

From Equation (1):

y = 3x + 4   (3)

Substitute y = 3x + 4 into Equation (2):

5x - 2(3x + 4) = -9

Simplify:

-x = -1 , x = 1

Substitute x = 1 into Equation (3):

y = 3(1) + 4 = 7

Thus, the fraction is: 17


Q8: Check graphically whether the pair of equations

3x + 5y = 15

x - y = 5

is consistent. Also, find the coordinates of the points where the graphs of equations meet the y-axis.

Sol: We have

3x + 5y = 15

 y =  15 - 3x5

Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

And from x - y = 5

⇒ y = x - 5

Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

Plotting the above two sets of points we get two straight lines l1 and l2 which intersect at the point (5, 0).

Thus, the given system is consistent.

Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

Obviously, line l1 meets the y-axis at (0, 3) and line l2 meets the y-axis at (0, - 5).

Q9: Places A and B are 160 km apart on the highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 8 hours, but if they travel towards each other, they meet in 2 hours. What are the speeds of the two cars? 

Sol: Let the car-I and car-II start from A and B at x km/hr and y km/hr respectively.

Case-I: [Cars are moving in the same direction]

Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

Let the two cars meet at C after 8 hours.

Distance covered:

by car-I = AC = 8x km

by car-II = BC = 8y km

∴ AB = AC - BC

⇒ 160 = 8x - 8y

⇒ x - y = 20   ...(1)

Case-II: [Cars are moving in opposite directions]

Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

Let, after 2 hours, the cars meet at D.
∴ Distance cover after 2 hours,
by car-I = AD = 2x km
by car-II = BD = 2y km
⇒ AB = AD + BD
⇒ 160 = 2x + 2y
⇒ 80 = x + y
⇒ x + y = 80 ...(2)
Adding (1) and (2), we get
x + y = 80
x - y = 20
2x = 100
⇒ x = 100/2 = 50
⇒ Substituting x = 50 in (1), we get
x - y = 20 ⇒ 50 - y = 20
⇒ y = 50 - 20 = 30
⇒ Speed of car-I = 50 km/hr
Speed of car-II = 30 km/hr.

Q10: Solve for x and y:

axb - bya =  a + b 

ax - by = 2ab 
Sol: We have:
= axbbya = a + b ...(1)
ax − by = 2ab    ...(2)
Dividing (2) by a, we have:

x -  bya = 2b 

⇒  bya =  x - 2b  ... (3)

From (1) and (3), we have:

axb - (x - 2b) =  a + b 

axb -  x + 2b - a - b = 0 

axb - x + b - a = 0

⇒ x ( ab  - 1 ) = - (b - a)

⇒ x ( a - bb ) = - (b - a)

⇒ x = - (b - a) x  b(a - b)

 ⇒ x = - (b - a) x  b- (b - a) = b

From (2),

ab - by = 2ab

⇒ - by = 2ab - ab = ab

-y =  abb =  a 

y = - a

Thus, x = b and y = - a.

Q11: The sum of two numbers is 8. Determine the numbers if the sum of their reciprocals is 8/15.

Sol: Let the two numbers be x and y.

According to the conditions:

x + y = 8 ...(1)

=  1x + 1y = 815 ... (2)

From (1), x = (8 - y)

Substituting x = (8 - y) in (2),

1(8 - y) + 1y = 815

⇒ y + 8 - y(8 - y)y = 815

⇒ 8 × 15 = 8 × y (8 - y)

⇒ 64y - 8y2 - 120 = 0

⇒- y2 + 8y - 15 = 0

⇒ y2 - 8y + 15 = 0

⇒ y2 - 5y - 3y + 15 = 0

⇒ y (y - 5) - 3 (y - 5) = 0

⇒ (y - 5) (y - 3) = 0

⇒ If y - 5 = 0 then y = 5

or if y - 3 = 0, then y = 3

Since x = 8 - y

⇒ when y = 5,

then x = 8 - 5 = 3

when y = 3, then

x = 8 - 3 = 5

⇒ The required numbers are (3, 5) or

(5, 3).

Q12: Solve the following pair of equations:

5x - 1 + 1y - 2 =  2 

6x - 1 - 3y - 2 =  1 

Sol: Let p = 1x - 1  and q = 1y - 2  

⇒The given system of equations becomes:

5p + q = 2 ...(1)

6p - 3q = 1 ...(2)

Multiplying (1) by 3 and adding to (2),

Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

 ⇒21p = 7 

⇒p = 721 = 13 

⇒ p =  13

and 5p + q = 2 

 ⇒ 5(13) + q = 2

⇒  53 + q = 2

⇒ q = 2 - 53 = 13 

⇒ q =  13 

Since, p = 1x - 1

therefore 1x - 1 = 13  ⇒  x - 1 = 3
⇒ x = 4
Also, q = 1y - 2
⇒ y - 2 = 3 ⇒ y = 5

Q13: Solve the following pair of equations:
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Sol: Let Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
⇒ The given pair of equation is expressed as
10p + 2q = 4 ⇒ 5p + q = 2   ...(1)
15p - 5q = - 2 ...(2)
Multiplying (1) by 5 and adding to (2)
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
From (1), Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
⇒1 + q = 2 ⇒ q = 2 - 1 = 1
Since, Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
⇒ x + y = 5   ...(3)
And Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
⇒ x − y = 1   ...(4)
Adding (3) and (4),
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
From (3), 3 + y = 5 ⇒ y = 2
Thus, x = 3 and y = 2.

Ques 14: Solve for x and y:
37x + 43y = 123
43x + 37y = 117
Sol: We have:
37x + 43y = 123 ...(1)
43x + 37y = 117 ...(2)
Adding (1) and (2)
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Dividing both sides by 80, we get
x + y = 3 ...(3)
Subtracting (2) from (1),
- 6x + 6y = 6 ⇒ - x + y = 1 ...(4)
Adding:
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
⇒ y = 4/2 = 2
Putting y = 2 in x + y = 3, we get
x + 2 = 3 ⇒ x = 3 - 2 = 1
Thus, x = 1 and y = 2.

Ques 15: Solve for ‘x’ and ‘y’:
(a - b) x + (a + b) y = a2 - 2ab - b2
(a + b) (x + y) = a2 + b2
Sol: We have:
(a - b) x + (a + b) y = a2 - 2ab - b2 ...(1)
(a + b) x + (a + b) y = a2 + b2   ...(2)
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
− 2b x = − 2ab − b
⇒ (− 2b) x = − 2b (a + b)
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
From (2),
(a + b) (a + b) + (a + b) y = a2 + b2
⇒(a + b)2 + (a + b) y = (a2 + b2)
⇒ (a + b) y = (a2 + b2) - (a + b)2
⇒ (a + b) y = a2 + b2 - (a2 + b2 + 2ab)
⇒ (a + b) y = a2 + b2 - a2 - b2 - 2ab
⇒ (a + b) y = - 2ab
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Thus, x = (a + b) and
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

Q16: Represent the following pair of equations graphically and write the co-ordinates of points where the lines intersect y-axis:
x + 3y = 6, 2x - 3y = 12
Sol: We have:
x + 3y = 6
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

and 2x - 3y = 12
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Plotting the above points, we get two straight lines l1 and l2 such that they intersect at (6, 0) as shown below:
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Obviously,
The line l1 meets the y-axis at (0, 2).
The line l2 meets the y-axis at (0, - 4).

Q17: Solve for x and y:

5x - 1 + 1y - 2 = 2

6x - 1 - 3y - 2 = 1

Sol: Given equations:

5x - 1 + 1y - 2 = 2

6x - 1 - 3y - 2 = 1

Substituting u = 1x - 1 and v = 1y - 2, we get:

5u + v = 2

6u - 3v = 1

Solving these equations:

From 5u + v = 2:
v = 2 - 5u

Substituting v = 2 - 5u into 6u - 3v = 1:
6u - 3(2 - 5u) = 1
6u - 6 + 15u = 1
21u = 7
u = 13

Finding v:
v = 2 - 5 × 13 = 13

Back-substituting to find x and y:

u = 1x - 1 ⇒ x - 1 = 3 ⇒ x = 4

v = 1y - 2 ⇒ y - 2 = 3 ⇒ y = 5

Solution:
x = 4, y = 5

Q18: For what values of ‘a’ and ‘b’ does the following pair of equations have an infinite number of solutions?
2x + 3y = 7
a (x + y) - b (x - y) = 3a + b - 2
Sol: We have:
2x + 3y = 7   ...(1)
a (x + y) - b (x - y) = 3a + b - 2   ...(2)
From (2), we have:
a (x + y) - b (x - y) = 3a + b - 2
⇒ ax + ay - bx + by = 3a + b - 2
⇒ ax - bx + ay + by = 3a + b - 2
⇒ (a - b) x + (a + b) y = 3a + b - 2
Now, A1 = 2, B1= 3,
C1 = - 7
A2 = (a - b), B2
= (a + b),
C2 = - [3a + b - 2]
For infinite number of solutions,
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
i.e., Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

⇒ 2 (a + b) = 3 (a − b)
⇒ 2a + 2b − 3a + 3b = 0
⇒ − a + 5b = 0
⇒ a = 5b ...(3)
Also = Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
⇒ 3 (3a + b - 2) = 7 (a + b)
⇒ 9a + 3b - 6 = 7a + 7b
⇒ 9a - 7a + 3b - 7b = 6
⇒ 2a - 4b = 6
⇒ a - 2b = 3    ...(4)
From (3) and (4),
5b - 2b = 3
⇒ 3b = 3 ⇒ b = 1
Thus, a = 5 × b
⇒ a = 5 × 1 = 5
i.e., a = 5 and b = 1.

Q19: Solve the following pairs of equations for x and y:
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

15x - y + 22x + y = 5

40x - y + 55x + y = 13 [x ≠ y and x ≠ -y]


Sol:  We are given the equations:

15x - y + 22x + y = 5

40x - y55x + y = 13

Let us substitute:

u = 1x - y, v = 1x + y

The equations become:

15u + 22v = 5

40u + 55v = 13

From the first equation:

u = 5 - 22v15

Substitute this value into the second equation:

40 × 5 - 22v15 + 55v = 13

Multiply through by 15:

200 - 880v + 825v = 195

Combine like terms:

200 - 55v = 195

Simplify:

-55v = -5 → v = 111

Substitute v back into the first equation:

15u + 22 × 111 = 5

Simplify:

15u + 2 = 5 → 15u = 3 → u = 15

Now back-substitute:

x - y = 5, x + y = 11

Add the equations:

2x = 16 → x = 8

Subtract the equations:

2y = 6 → y = 3

Final Answer: x = 8, y = 3

Q20: Draw the graph of the pair of linear equations x – y + 2 = 0 and 4x – y – 4 = 0. Calculate the area of triangle formed by the lines so drawn and the x-axis. 
Sol: 
To draw the graph of the given pair of equations, we have the table of ordered pairs
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Plot the points A(0, 2), B(–2, 0); C(0, –4) and D(1, 0) on the graph paper and join the points to form the lines AB and CD :
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
From the graph, we find that the points P(2, 4) is common to both the lines AB and CD.
These lines meet the x-axis at B(–2, 0) and D(1, 0).
Thus, the triangle BDP is formed by the lines and the x-axis.
The vertices of this Δ are
B(–2, 0), D(1, 0)  and    P(2, 4)
Now, the base of ΔBDP = BD
= (BO + OD)
= (2 + 1) units
= 3 units
Altitude of the ΔBDP = PQ
= 4 units
∴ Area of ΔBDP = Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
=  6 sq. units

Q21: It can take 12 hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for 4 hours and the pipe of smaller diameter for 9 hours, only half the pool can be filled. How long would it take for each pipe to fill the pool separately?
Sol: Let the time taken by the pipe of larger diameter to fill the pool separately = x hours.
The time taken by the pipe of smaller diameter to fill the pool separately = y hours.
∴ Part of the pool fill by a pipe of larger diameter in 1 hour = 1/x.
Part of fool filled by the pipe of larger diameter in 4 hours = 4/x.
Similarly, Part of the pool filled by the pipe of smaller diameter in 9 hours = 9/y.
∴We have =  Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables...(1)
Since the pool is filled by both the pipes together in 12 hours.
∴  Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables ...(2)
To solve (1) and (2), multiplying (1) by 3 and subtracting (2) from it, we have
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Substituting, y = 30 in (2), we have
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
⇒ x = 20
⇒ Required time taken by pipe of larger diameter = 20 hours

Required time taken by pipe of smaller diameter = 30 hours

The document Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables is a part of the Class 10 Course Mathematics (Maths) Class 10.
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FAQs on Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

1. What are the different methods to solve a pair of linear equations in two variables?
Ans. There are three main methods to solve a pair of linear equations in two variables: 1. Graphical Method: This involves plotting the equations on a graph to find the point of intersection, which represents the solution. 2. Substitution Method: In this method, one equation is solved for one variable, and then that expression is substituted into the other equation. 3. Elimination Method: This method involves adding or subtracting the equations to eliminate one variable, making it easier to solve for the other variable.
2. How do you determine if a pair of linear equations is consistent or inconsistent?
Ans. A pair of linear equations is considered consistent if they have at least one solution, meaning the lines intersect at one point (for intersecting lines) or are identical (for coincident lines). They are inconsistent if there is no solution, which occurs when the lines are parallel and never intersect. This can be determined by comparing the ratios of the coefficients of the variables in the equations.
3. What is the geometric interpretation of a pair of linear equations in two variables?
Ans. The geometric interpretation of a pair of linear equations in two variables involves representing each equation as a line on a Cartesian plane. The point where the two lines intersect represents the solution to the equations. If the lines are parallel, there is no solution (inconsistent). If they overlap, they represent the same line, indicating infinite solutions (coincident).
4. Can you explain how to apply the substitution method with an example?
Ans. Yes! To apply the substitution method, follow these steps: 1. Solve one of the equations for one variable. 2. Substitute this expression into the other equation. For example, consider the equations: 1) \(x + y = 10\) 2) \(2x - y = 3\) First, solve the first equation for \(y\): \(y = 10 - x\). Then, substitute \(y\) into the second equation: \(2x - (10 - x) = 3\). Solving this gives \(3x - 10 = 3\) or \(3x = 13\), so \(x = \frac{13}{3}\). Substituting back to find \(y\), we get \(y = 10 - \frac{13}{3} = \frac{17}{3}\). The solution is \((\frac{13}{3}, \frac{17}{3})\).
5. What are some real-life applications of solving pairs of linear equations in two variables?
Ans. Pairs of linear equations in two variables are used in various real-life situations, such as: 1. Budgeting: Determining the costs of two items and finding the possible combinations within a budget. 2. Mixture Problems: Figuring out the proportions of different solutions or substances needed to achieve a desired mixture. 3. Supply and Demand: Analyzing how changes in price affect the quantity supplied and quantity demanded in markets. These applications demonstrate how linear equations can model and solve practical problems.
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