RS Aggarwal Solutions Class 10 Chapter 3 Linear equations in two variables EX 3D

In this chapter, we provide RS Aggarwal Solutions for Class 10 Chapter 3 Linear equations in two variables EX 3D Maths for English medium students, Which will very helpful for every student in their exams. Students can download the latest RS Aggarwal Solutions for Class 10 Chapter 3 Linear equations in two variables EX 3D Maths pdf, free RS Aggarwal Solutions Class 10 Chapter 3 Linear equations in two variables EX 3D Maths book pdf download. Now you will get step by step solution to each question.

TextbookNCERT
ClassClass 10
SubjectMaths
ChapterChapter 3
Chapter NameLinear equations in two variables
Exercise3 D
CategoryRS Aggarwal Solutions

RS Aggarwal Solutions Class 10 Chapter 3 Linear equations in two variables Ex 3D

Question 1:

RS Aggarwal Solutions Class 10 Chapter 3 Linear equations in two variables ex 3d

Question 2:

RS Aggarwal Solutions Class 10 Chapter 3 Linear equations in two variables ex 3d

Question 3:
3x – 5y – 7 = 0
6x – 10y – 3 = 0

RS Aggarwal Solutions Class 10 Chapter 3 Linear equations in two variables ex 3d

Hence the given system of equations is inconsistent
Question 4:
2x – 3y – 5 = 0, 6x – 9y – 15 = 0
These equations are of the form

RS Aggarwal Solutions Class 10 Chapter 3 Linear equations in two variables ex 3d

Hence the given system of equations has infinitely many solutions
Question 5:
kx + 2y – 5 = 0
3x – 4y – 10 = 0
These equations are of the form

RS Aggarwal Class 10 Solutions Chapter 3 Linear equations in two variables ex 3d

This happens when
(kneq frac { -3 }{ 2 })
Thus, for all real value of k other that , the given system equations will have a unique solution
(ii) For no solution we must have

RS Aggarwal Class 10 Solutions Chapter 3 Linear equations in two variables ex 3d
RS Aggarwal Class 10 Solutions Chapter 3 Linear equations in two variables ex 3d

Hence, the given system of equations has no solution if (k=frac { -3 }{ 2 } )
Question 6:
x + 2y – 5 = 0
3x + ky + 15 = 0
These equations are of the form of

RS Aggarwal Class 10 Solutions Chapter 3 Linear equations in two variables ex 3d

Thus for all real value of k other than 6, the given system of equation will have unique solution
(ii) For no solution we must have

Solution of RS Aggarwal Class 10 Chapter 3 Linear equations in two variables ex 3d

Therefore k = 6
Hence the given system will have no solution when k = 6.
Question 7:
x + 2y – 3 = 0, 5x + ky + 7 = 0
These equations are of the form

Solution of RS Aggarwal Class 10 Chapter 3 Linear equations in two variables ex 3d

(i) For a unique solution we must have

Solution of RS Aggarwal Class 10 Chapter 3 Linear equations in two variables ex 3d

Thus, for all real value of k other than 10
The given system of equation will have a unique solution.
(ii) For no solution we must have

Solution of RS Aggarwal Class 10 Chapter 3 Linear equations in two variables ex 3d

Hence the given system of equations has no solution if

RS Aggarwal Solutions Class 10 2017 Chapter 3 Linear equations in two variables ex 3d

For infinite number of solutions we must have

RS Aggarwal Solutions Class 10 2017 Chapter 3 Linear equations in two variables ex 3d

This is never possible since

RS Aggarwal Solutions Class 10 2017 Chapter 3 Linear equations in two variables ex 3d

There is no value of k for which system of equations has infinitely many solutions
Question 8:
8x + 5y – 9 = 0
kx + 10y – 15 = 0
These equations are of the form

rRS Aggarwal Solutions Class 10 2017 Chapter 3 Linear equations in two variables ex 3d

Clearly, k = 16 also satisfies the condition

RS Aggarwal Solutions Class 10 2017 Chapter 3 Linear equations in two variables ex 3d

Hence, the given system will have no solution when k = 16.
Question 9:
kx + 3y – 3 = 0 —-(1)
12x + ky – 6 = 0 —(2)

Class 10 RS Aggarwal Solutions Chapter 3 Linear equations in two variables ex 3d

These equations are of the form

Class 10 RS Aggarwal Solutions Chapter 3 Linear equations in two variables ex 3d

Hence, the given system will have no solution when k = -6
Question 10:
3x + y – 1 = 0
(2k – 1)x + (k – 1)y – (2k + 1) = 0
These equations are of the form

Class 10 RS Aggarwal Solutions Chapter 3 Linear equations in two variables ex 3d

Thus,

Class 10 RS Aggarwal Solutions Chapter 3 Linear equations in two variables ex 3d

Hence the given equation has no solution when k = 2
Question 11:
(3k + 1)x + 3y – 2 = 0

Class 10 RS Aggarwal Solutions Chapter 3 Linear equations in two variables ex 3d

these equations are of the form

RS Aggarwal Class 10 Maths Book Solutions pdf Chapter 3 Linear equations in two variables ex 3d

Thus, k = -1 also satisfy the condition

RS Aggarwal Class 10 Maths Book Solutions pdf Chapter 3 Linear equations in two variables ex 3d

Hence, the given system will have no solution when k = -1
Question 12:
The given equations are
3x – y – 5 = 0 —(1)
6x – 2y + k = 0—(2)

RS Aggarwal Class 10 Maths Book Solutions pdf Chapter 3 Linear equations in two variables ex 3d

Equations (1) and (2) have no solution, if

RS Aggarwal Class 10 Maths Book Solutions pdf Chapter 3 Linear equations in two variables ex 3d

Question 13:
kx + 2y – 5 = 0
3x + y – 1 = 0
These equations are of the form

RS Aggarwal Class 10 Book pdf download Chapter 3 Linear equations in two variables ex 3d

Thus, for all real values of k other than 6, the given system of equations will have a unique solution
Question 14:
x – 2y – 3 = 0
3x + ky – 1 = 0
These equations are of the form of

RS Aggarwal Class 10 Book pdf download Chapter 3 Linear equations in two variables ex 3d

Thus, for all real value of k other than -6, the given system of equations will have a unique solution
Question 15:
kx + 3y – (k – 3) = 0
12x + ky – k = 0
These equations are of the form

RS Aggarwal Class 10 Book pdf download Chapter 3 Linear equations in two variables ex 3d

Thus, for all real value of k other than , the given system of equations will have a unique solution
Question 16:
4x – 5y – k = 0, 2x – 3y – 12 = 0
These equations are of the form

RS Aggarwal Class 10 Book pdf download Chapter 3 Linear equations in two variables ex 3d

Thus, for all real value of k the given system of equations will have a unique solution
Question 17:
2x + 3y – 7 = 0
(k – 1)x + (k + 2)y – 3k = 0
These are of the form

Class 10 Maths RS Aggarwal Solutions Chapter 3 Linear equations in two variables ex 3d

This hold only when

Class 10 Maths RS Aggarwal Solutions Chapter 3 Linear equations in two variables ex 3d

Now the following cases arises
Case : I

Class 10 Maths RS Aggarwal Solutions Chapter 3 Linear equations in two variables ex 3d

Case: II

Class 10 Maths RS Aggarwal Solutions Chapter 3 Linear equations in two variables ex 3d

Case III

Class 10 Maths RS Aggarwal Solutions Chapter 3 Linear equations in two variables ex 3d

For k = 7, there are infinitely many solutions of the given system of equations
Question 18:
2x + (k – 2)y – k = 0
6x + (2k – 1)y – (2k + 5) = 0
These are of the form

Maths RS Aggarwal Class 10 Solutions Chapter 3 Linear equations in two variables ex 3d

For infinite number of solutions, we have

Maths RS Aggarwal Class 10 Solutions Chapter 3 Linear equations in two variables ex 3d

This hold only when

Maths RS Aggarwal Class 10 Solutions Chapter 3 Linear equations in two variables ex 3d

Case (1)

Maths RS Aggarwal Class 10 Solutions Chapter 3 Linear equations in two variables ex 3d

Case (2)

Maths RS Aggarwal Class 10 Solutions Chapter 3 Linear equations in two variables ex 3d

Case (3)

RS Aggarwal Solutions Class 10 2018 Chapter 3 Linear equations in two variables ex 3d

Thus, for k = 5 there are infinitely many solutions
Question 19:
kx + 3y – (2k +1) = 0
2(k + 1)x + 9y – (7k + 1) = 0
These are of the form

RS Aggarwal Solutions Class 10 2018 Chapter 3 Linear equations in two variables ex 3d

For infinitely many solutions, we must have

RS Aggarwal Solutions Class 10 2018 Chapter 3 Linear equations in two variables ex 3d

This hold only when

RS Aggarwal Solutions Class 10 2018 Chapter 3 Linear equations in two variables ex 3d

Now, the following cases arise
Case – (1)

RS Aggarwal Solutions Class 10 2018 Chapter 3 Linear equations in two variables ex 3d

Case (2)

rs-aggarwal-class-10-solutions-linear-equations-in-two-variables-ex-3d-q19-5

Case (3)

rs-aggarwal-class-10-solutions-linear-equations-in-two-variables-ex-3d-q19-6
rs-aggarwal-class-10-solutions-linear-equations-in-two-variables-ex-3d-q19-7

Thus, k = 2, is the common value for which there are infinitely many solutions
Question 20:
5x + 2y – 2k = 0
2(k +1)x + ky – (3k + 4) = 0
These are of the form

rs-aggarwal-class-10-solutions-linear-equations-in-two-variables-ex-3d-q20-1

For infinitely many solutions, we must have

rs-aggarwal-class-10-solutions-linear-equations-in-two-variables-ex-3d-q20-2

These hold only when

rs-aggarwal-class-10-solutions-linear-equations-in-two-variables-ex-3d-q20-3

Case I

rs-aggarwal-class-10-solutions-linear-equations-in-two-variables-ex-3d-q20-4
rs-aggarwal-class-10-solutions-linear-equations-in-two-variables-ex-3d-q20-5
rs-aggarwal-class-10-solutions-linear-equations-in-two-variables-ex-3d-q20-6

Thus, k = 4 is a common value for which there are infinitely by many solutions.
Question 21:
x + (k + 1)y – 5 = 0
(k + 1)x + 9y – (8k – 1) = 0
These are of the form

RS Aggarwal Solutions Class 10 Chapter 3 Linear equations in two variables ex 3d

For infinitely many solutions, we must have

RS Aggarwal Solutions Class 10 Chapter 3 Linear equations in two variables ex 3d

Question 22:
(k – 1)x – y – 5 = 0
(k + 1)x + (1 – k)y – (3k + 1) = 0
These are of the form

RS Aggarwal Solutions Class 10 Chapter 3 Linear equations in two variables ex 3d

For infinitely many solution, we must now

RS Aggarwal Solutions Class 10 Chapter 3 Linear equations in two variables ex 3d

k = 3 is common value for which the number of solutions is infinitely many
Question 23:
(a – 1)x + 3y – 2 = 0
6x + (1 – 2b)y – 6 = 0
These equations are of the form

RS Aggarwal Solutions Class 10 Chapter 3 Linear equations in two variables ex 3d

For infinite many solutions, we must have

RS Aggarwal Solutions Class 10 Chapter 3 Linear equations in two variables ex 3d

Hence a = 3 and b = -4
Question 24:
(2a – 1)x + 3y – 5 = 0
3x + (b – 1)y – 2 = 0
These equations are of the form

RS Aggarwal Class 10 Solutions Chapter 3 Linear equations in two variables ex 3d

These holds only when

RS Aggarwal Class 10 Solutions Chapter 3 Linear equations in two variables ex 3d

Question 25:
2x – 3y – 7 = 0
(a + b)x + (a + b – 3)y – (4a + b) = 0
These equation are of the form

RS Aggarwal Class 10 Solutions Chapter 3 Linear equations in two variables ex 3d

For infinite number of solution

RS Aggarwal Class 10 Solutions Chapter 3 Linear equations in two variables ex 3d

Putting a = 5b in (2), we get

RS Aggarwal Class 10 Solutions Chapter 3 Linear equations in two variables ex 3d

Putting b = -1 in (1), we get

Solution of RS Aggarwal Class 10 Chapter 3 Linear equations in two variables ex 3d

Thus, a = -5, b = -1
Question 27:
The given equations are
2x + 3y = 7 —-(1)
a(x + y) – b(x – y) = 3a + b – 2 —(2)
Equation (2) is
ax + ay – bx + by = 3a + b – 2
(a – b)x + (a + b)y = 3a + b -2
Comparing with the equations

Solution of RS Aggarwal Class 10 Chapter 3 Linear equations in two variables ex 3d

There are infinitely many solution

Solution of RS Aggarwal Class 10 Chapter 3 Linear equations in two variables ex 3d

2a + 2b = 3a – 3b and 3(3a + b – 2) = 7(a + b)
-a = -5b and 9a + 3b – 6 = 7a + 7b
a = 5b and 9a – 7a + 3b – 7b = 6
or 2a – 4b = 6
or a – 2b = 3
thus equation in a, b are
a = 5b —(3)
a – 2b = 3 —(4)
putting a = 5b in (4)
5b – 2b = 3 or 3b = 3 Þ b = 1
Putting b = 1 in (3)
a = 5 and b = 1
Question 28:
We have 5x – 3y = 0 —(1)
2x + ky = 0 —(2)
Comparing the equation with

RS Aggarwal Solutions Class 10 2017 Chapter 3 Linear equations in two variables ex 3d

These equations have a non – zero solution if

RS Aggarwal Solutions Class 10 2017 Chapter 3 Linear equations in two variables ex 3d
RS Aggarwal Solutions Class 10 2017 Chapter 3 Linear equations in two variables ex 3d

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