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,
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
⇒ p ≠ 2 × 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 opposite angles are supplementary.
∴∠P + ∠R = 180°
⇒ (2x + 4)° + (2y + 10)° = 180°
⇒ 2x + 2y + 14 - 180 = 0
⇒ 2x + 2y - 166 = 0
⇒ x + y - 83 = 0  ...(1)
Also ∠Q + ∠S = 180°
∴ (y + 3)° + (4x - 5)° = 180°
⇒ y + 4x - 2 - 180 = 0
⇒ y + 4x - 182 = 0 ...(2)
From (1) and (2),
a1 = 1, b1 = 1, c1 = - 83
a2 = 4, b2 = 1, c2 = - 182
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
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
∴ ∠P = (2x + 4)° = [(2 × 33) + 4]° = 70°
∠Q = (y + 3)° = [50 + 3]° = 53°
∠R = (2y + 10)° = [2 × 50 + 10]° = 110°
∠S = (4x - 5)° = [4 × 33 - 5°] = 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 58]
Subtracting (1) from (2),
35x + 23y = 197
23x + 35y = 209
(-) (-) (-)
12x - 12y = - 12
⇒ x - y = 1 ...(4) [Dividing by 12]
Adding (3) and (4),
2x = 8 ⇒ x = 4
From (3) 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: From (1) and (2), we have:
a1 = 3, b1 = 5, c1 = - 70
a2 = 7, b2 = - 3, c2 = - 60
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
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
Thus, the required solution is:
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables .

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, Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
∴We have
= Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
This condition represents parallel lines. Hence, the given pair represents parallel lines.

Q6: Check graphically whether the pair of equations 
3x - 2y + 2 = 0
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables= 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 x/y.
From 1st condition,
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

⇒ 3x + 6 = y + 2
⇒ 3x − y + 4 = 0 ...(1)
From 2nd conditon,
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

⇒ 5x + 15 = 2y + 6
⇒ 5x − 2y + 9 = 0 ...(2)
From (1) and (2), we have:
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

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
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
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:
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Sol: We have:
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables ...(1)
ax − by = 2ab    ...(2)
Dividing (2) by a, 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 ...(3)
From (1) and (3), 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
From (2),
ab - by = 2ab
⇒ - by = 2ab - ab = ab
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
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)
=Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables ..(2)
From (1), x = (8 - y)
Substituting x = (8 - y) in (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
⇒ 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:
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 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
and 5p + q = 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
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

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
= 3 ⇒ x = 4
Also, Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
⇒ y - 2 = 3 ⇒ y = 5
Thus, x = 4 and y = 4
⇒ 3y - 6 = 1
⇒ 3y = 1 + 6 = 7
⇒ y = 7/3

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:
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
∴ We have:
5p + q = 2   ...(1)
6p - 3q = 1   ...(2)
From (1) and (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
Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
And Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
Now, Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
⇒ x − 1 = 3 ⇒ x = 4
And Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
⇒ y − 2 = 3 ⇒ y = 5
Thus x = 4 and 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
Sol: Let Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables
∴ We have:
15p + 22q = 5 ...(1)
40p + 55q = 13 ...(2)
From (1) and (2), we get
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
Adding (3) and (4), we have
2x = 16 ⇒ x = 16/2 =8
From (4), 8 + y = 11 ⇒ y = 11 - 8 = 3
Thus, x = 8 and 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. <b>Graphical Method</b>: This involves plotting the equations on a graph to find the point of intersection, which represents the solution. 2. <b>Substitution Method</b>: In this method, you solve one equation for one variable and substitute that expression into the other equation. 3. <b>Elimination Method</b>: Here, you manipulate the equations to eliminate one variable, allowing you to solve for the other variable directly.
2. How can we represent a pair of linear equations graphically ?
Ans. A pair of linear equations can be represented graphically by plotting each equation as a straight line on a coordinate plane. The point where the two lines intersect represents the solution to the equations. If the lines are parallel, there is no solution (inconsistent system). If the lines coincide, there are infinitely many solutions (dependent system).
3. What is the significance of the coefficients in a pair of linear equations ?
Ans. The coefficients in a pair of linear equations determine the slope and position of the lines represented by the equations. They affect the steepness of the line and how the two lines relate to each other (whether they are parallel, intersecting, or coincident). This relationship helps in understanding the nature of the solutions.
4. Can a pair of linear equations have no solution ? If so, how ?
Ans. Yes, a pair of linear equations can have no solution when the lines represented by the equations are parallel. This occurs when the equations have the same slope but different y-intercepts. In such cases, there is no point of intersection, indicating that the equations are inconsistent.
5. How do you verify the solution of a pair of linear equations ?
Ans. To verify the solution of a pair of linear equations, you substitute the values of the variables obtained from solving the equations back into both original equations. If both equations hold true with these values, then the solution is verified. If either equation does not hold true, the solution is incorrect.
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