Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set 1.3

Maharashtra state Board 10th Standard Solutions Chapter 1 Linear Equations in two Variables – Here are all the MH Board solutions for 10th Standard maths Practice Set 1.3. This solution contains questions, answers, images, explanations of the complete Practice Set 1.3 titled Linear Equations in two Variables of maths taught in 10th Standard. If you are a student of 10th Standard who is using Maharashtra state Board Textbook to study maths, then you must come across Practice Set 1.3 Linear Equations in two Variables. After you have studied lesson, you must be looking for answers of its questions. Here you can get complete Maharashtra Board Solutions for 10th Standard maths Chapter 1 Linear Equations in two Variables in one place.

Maharashtra State Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set 1.3

Question 1.
Fill in the blanks with correct number.
Solution:
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set Ex 1.3 1

Question 2.
Find the values of following determinants.
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set Ex 1.3 2
Solution:
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set Ex 1.3 3
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set Ex 1.3 4

Question 3.
Solve the following simultaneous equations using Cramer’s rule.
i. 3x – 4y = 10 ; 4x + 3y = 5
ii. 4x + 3y – 4 = 0 ; 6x = 8 – 5y
iii. x + 2y = -1 ; 2x – 3y = 12
iv. 6x – 4y = -12 ; 8x – 3y = -2
v. 4m + 6n = 54 ; 3m + 2n = 28
vi. 2x + 3y = 2 ; x – \(\frac { y }{ 2 } \) = \(\frac { 1 }{ 2 } \)
Solution:
i. The given simultaneous equations are 3x – 4y = 10 …(i)
4x + 3y = 5 …(ii)
Equations (i) and (ii) are in ax + by = c form.
Comparing the given equations with
a1x + b1y = c1 and a2x + b2y = c2, we get
a1 = 3, b1 = -4, c1 = 10 and
a2 = 4, b2 = 3, c2 = 5
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set Ex 1.3 5
∴ (x, y) = (2, -1) is the solution of the given simultaneous equations.

ii. The given simultaneous equations are
4x + 3y – 4 = 0
∴ 4x + 3y = 4 …(i)
6x = 8 – 5y
∴ 6x + 5y = 8 …(ii)
Equations (i) and (ii) are in ax + by = c form.
Comparing the given equations with
a1x + b1y = c1 and a2x + b2y = c2, we get
a1 = 4, b1 = 3, c1 = 4 and
a2 = 6, b2 = 5, c2 = 8
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set Ex 1.3 6
∴ (x, y) = (-2, 4) is the solution of the given simultaneous equations.

iii. The given simultaneous equations are
x + 2y = -1 …(i)
2x – 3y = 12 …(ii)
Equations (i) and (ii) are in ax + by = c form.
Comparing the given equations with
a1x + b1y = C1 and a2x + b2y = c2, we get
a1 = 1, b1 = 2, c1 = -1 and
a2 = 2, b2 = -3, c2 = 12
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set Ex 1.3 7
∴ (x, y) = (3, -2) is the solution of the given simultaneous equations.

iv. The given simultaneous equations are
6x – 4y = -12
∴ 3x – 2y = -6 …(i) [Dividing both sides by 2]
8x – 3y = -2 …(ii)
Equations (i) and (ii) are in ax + by = c form.
Comparing the given equations with
a1x + b1y = c1 and a2x + b2y = c2, we get
a1 = 3, b1 = -2, c1 = -6 and
a2 = 8, b2 = -3, c2 = -2
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set Ex 1.3 8
∴ (x, y) = (2, 6) is the solution of the given simultaneous equations.

v. The given simultaneous equations are
4m + 6n = 54
2m + 3n = 27 …(i) [Dividing both sides by 2]
3m + 2n = 28 …(ii)
Equations (i) and (ii) are in am + bn = c form.
Comparing the given equations with
a1m + b1n = c1 and a2m + b2n = c2, we get
a1 = 2, b1 = 3, c1 = 27 and
a2 = 3, b2 = 2, c2 = 28
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set Ex 1.3 9
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set Ex 1.3 10
∴ (m, n) = (6, 5) is the solution of the given simultaneous equations.

vi. The given simultaneous equations are
2x + 3y = 2 …(i)
x = \(\frac { y }{ 2 } \) = \(\frac { 1 }{ 2 } \)
∴ 2x – y = 1 …(ii) [Multiplying both sides by 2]
Equations (i) and (ii) are in ax + by = c form.
Comparing the given equations with
a1x + b1y = c1 and a2x + b2y = c2, we get
a1 = 2, b1 = 3, c1 = 2 and
a2 = 2, b2 = -1, c2 = 1
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set Ex 1.3 11

Question 1.
To solve the simultaneous equations by determinant method, fill in the blanks,
y + 2x – 19 = 0; 2x – 3y + 3 = 0 (Textbookpg.no. 14)
Solution:
Write the given equations in the form
ax + by = c.
2x + y = 19
2x – 3y = -3
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set Ex 1.3 12

Question 2.
Complete the following activity. (Textbook pg. no. 15)
Solution:
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set Ex 1.3 13

Question 3.
What is the nature of solution if D = 0? (Textbook pg. no. 16)
Solution:
If D = 0, i.e. a1b2 – b1a2 = 0, then the two simultaneous equations do not have a unique solution.
Examples:
i. 2x – 4y = 8 and x – 2y = 4
Here, a1b2 – b1a2 = (2)(-2) – (-4) (1)
= -4 + 4 = 0
Graphically, we can check that these two lines coincide and hence will have infinite solutions.

ii. 2x – y = -1 and 2x – y = -4
Here, a1 b2 – b1 a2 = (2)(-1) – (-1) (2)
= -2 + 2 = 0
Graphically, we can check that these two lines are parallel and hence they do not have a solution.

Question 4.
What can you say about lines if common solution is not possible? (Textbook pg. no. 16)
Answer:
If the common solution is not possible, then the lines will either coincide or will be parallel to each other.

Maharashtra Board Class 10 Maths Solutions

Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set 1.2

Maharashtra state Board 10th Standard Solutions Chapter 1 Linear Equations in two Variables – Here are all the MH Board solutions for 10th Standard maths Practice Set 1.2. This solution contains questions, answers, images, explanations of the complete Practice Set 1.2 titled Linear Equations in two Variables of maths taught in 10th Standard. If you are a student of 10th Standard who is using Maharashtra state Board Textbook to study maths, then you must come across Practice Set 1.2 Linear Equations in two Variables. After you have studied lesson, you must be looking for answers of its questions. Here you can get complete Maharashtra Board Solutions for 10th Standard maths Chapter 1 Linear Equations in two Variables in one place.

Maharashtra State Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set 1.2

Question 1.
Complete the following table to draw graph of the equations.
i. x + y = 3
ii. x – y = 4
Answer:
i. x + y = 3
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 1
ii. x – y = 4
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 2

Question 2.
Solve the following simultaneous equations graphically.
i. x + y = 6 ; x – y = 4
ii. x + y = 5 ; x – y = 3
iii. x + y = 0 ; 2x – y = 9
iv. 3x – y = 2 ; 2x – y = 3
v. 3x – 4y = -7 ; 5x – 2y = 0
vi. 2x – 3y = 4 ; 3y – x = 4
Solution:
i. The given simultaneous equations are
x + y = 6                                                                                                        x – y = 4
∴ y = 6 – x                                                                                                     ∴ y = x – 4Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 3
The two lines intersect at point (5, 1).
∴ x = 5 and y = 1 is the solution of the simultaneous equations x + y = 6 and x – y = 4.

ii. The given simultaneous equations are
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 6
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 7
The two lines intersect at point (4, 1).
∴ x = 4 and y = 1 is the solution of the simultaneous equations x+y = 5 and x – y = 3.

iii. The given simultaneous equations are
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 4
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 5
The two lines intersect at point (3, -3).
∴ x = 3 and y = -3 is the solution of the simultaneous equations x + y = 0 and 2x – y = 9.

iv. The given simultaneous equations are
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 8
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 9
The two lines intersect at point (-1, -5).
∴ x = -1 and y = -5 is the solution of the simultaneous equations 3x- y = 2 and 2x- y = 3.

v. The given simultaneous equations are
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 10
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 11
The two lines intersect at point (1, 2.5).
∴ x = 1 and y = 2.5 is the solution of the simultaneous equations 3x – 4y = -7 and 5x – 2y = 0.

vi. The given simultaneous equations are
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 12
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 13
The two lines intersect at point (8, 4).
∴ x = 8 and y = 4 is the solution of the simultaneous equations 2x – 3y = 4 and 3y – x = 4.

Question 1.
Solve the following simultaneous equations by graphical method. Complete the following tables to get ordered pairs.
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 14
i. Plot the above ordered pairs on the same co-ordinate plane.
ii. Draw graphs of the equations.
iii. Note the co-ordinates of the point of intersection of the two graphs. Write solution of these equations. (Textbook pg. no. 8)
Solution:
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 15 Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 16
The two lines intersect at point (-1, -2).
∴ (x , y) = (-1, -2) is the solution of the given simultaneous equations.

Question 1.
Solve the above equations by method of elimination. Check your solution with the solution obtained by graphical method. (Textbook pg. no. 8)
Solution:
The given simultaneous equations are
x – y = 1 …(i)
5x – 3y = 1 …(ii)
Multiplying equation (i) by 3, we get
3x – 3y = 3 …(iii)
Subtracting equation (iii) from (ii), we get
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 17
Substituting x = -1 in equation (i), we get
-1 -y= 1
∴ -y = 1 + 1
∴ -y = 2
∴ y = -2
∴ (x,y) = (-1, -2) is the solution of the given simultaneous equations.
∴ The solution obtained by elimination method and by graphical method is the same.

Question 2.
The following table contains the values of x and y co-ordinates for ordered pairs to draw the graph of 5x – 3y = 1.
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 18
i. Is it easy to plot these points?
ii. Which precaution is to be taken to find ordered pairs so that plotting of points becomes easy? (Textbook pg. no. 8)
Solution:
i. No
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 19
The above numbers are non-terminating and recurring decimals.
∴ It is not easy to plot the given points.

ii. While finding ordered pairs, numbers should be selected in such a way that the co-ordinates obtained will be integers.

Question 3.
To solve simultaneous equations x + 2y = 4; 3x + 6y = 12 graphically, following are the ordered pairs.
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 20
Plotting the above ordered pairs, graph is drawn. Observe it and find answers of the following questions.
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 21
i. Are the graphs of both the equations different or same?
ii. What are the solutions of the two equations x + 2y = 4 and 3x + 6y = 12? How many solutions are possible?
iii. What are the relations between coefficients of x, coefficients of y and constant terms in both the equations?
iv. What conclusion can you draw when two equations are given but the graph is only one line? (Textbook pg. no. 9)
Solution:
i. The graphs of both the equations are same.
ii. The solutions of the given equations are (-2, 3), (0, 2), (1, 1.5), etc.
∴ Infinite solutions are possible.
iii. Ratio of coefficients of x = \(\frac { 1 }{ 3 } \)
Ratio of coefficients of y = \(\frac { 2 }{ 6 } \) = \(\frac { 1 }{ 3 } \)
Ratio of constant terms = \(\frac { 4 }{ 12 } \) = \(\frac { 1 }{ 3 } \)
∴ Ratios of coefficients of x = ratio of coefficients of y = ratio of the constant terms
iv. When two equations are given but the graph is only one line, the equations will have infinite solutions.

Question 4.
Draw graphs of x- 2y = 4, 2x – 4y = 12 on the same co-ordinate plane. Observe it. Think of the relation between the coefficients of x, coefficients ofy and the constant terms and draw the inference. (Textbook pg. no. 10)
Solution:
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 22 Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 23
ii. Ratio of coefficients of x =\(\frac { 1 }{ 2 } \)
Ratio of coefficients of y = \(\frac { -2 }{ -4 } \) = \(\frac { 1 }{ 2 } \)
Ratio of constant terms = \(\frac { 4 }{ 12 } \) = \(\frac { 1 }{ 3 } \)
∴ Ratio of coefficients of x = ratio of coefficients of y ratio of constant terms
iii. If ratio of coefficients of x = ratio of coefficients of y ≠ ratio of constant terms, then the graphs of the two equations will be parallel to each other.

Condition of consistency in Equations:
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.2 24

Maharashtra Board Class 10 Maths Solutions

Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set 1.1

Maharashtra state Board 10th Standard Solutions Chapter 1 Linear Equations in two Variables – Here are all the MH Board solutions for 10th Standard maths Practice Set 1.1. This solution contains questions, answers, images, explanations of the complete Practice Set 1.1 titled Linear Equations in two Variables of maths taught in 10th Standard. If you are a student of 10th Standard who is using Maharashtra state Board Textbook to study maths, then you must come across Practice Set 1.1 Linear Equations in two Variables. After you have studied lesson, you must be looking for answers of its questions. Here you can get complete Maharashtra Board Solutions for 10th Standard maths Chapter 1 Linear Equations in two Variables in one place.

Maharashtra State Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Practice Set 1.1

Question 1.
Complete the following activity to solve the simultaneous equations.
5x + 3y = 9 …(i)
2x-3y=12 …(ii)
Solution:
5x + 3y = 9 …(i)
2x-3y=12 …(ii)
Add equations (i) and (ii).
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.1 1

Question 2.
Solve the following simultaneous equations.
i. 3a + 5b = 26; a + 5b = 22
ii. x + 7y = 10; 3x – 2y = 7
iii. 2x – 3y = 9; 2x + y = 13
iv. 5m – 3n = 19; m – 6n = -7
v. 5x + 2y = -3;x + 5y = 4
vi. \(\frac { 1 }{ 3 } \) x+ y = \(\frac { 10 }{ 3 } \) ; 2x + \(\frac { 1 }{ 4 } \) y = \(\frac { 11 }{ 4 } \)
vii. 99x + 101y = 499 ; 101x + 99y = 501
viii. 49x – 57y = 172; 57x – 49y = 252
Solution:
i. 3a + 5b = 26 …(i)
a + 5b = 22 …(ii)
Subtracting equation (ii) from (i), we get
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.1 2
Substituting a = 2 in equation (ii), we get
2 + 5b = 22
∴ 5b = 22 – 2
∴ 5b = 20
∴ b = \(\frac { 20 }{ 5 } \) =4
∴ (a, b) = (2, 4) is the solution of the given simultaneous equations.

ii. x + 7y = 10
∴ x = 10 – 7y …(i)
3x – 2y = 7 …1(ii)
Substituting x = 10 – ly in equation (ii), we get
3 (10 – 7y) – 2y = 7
∴ 30 – 21y – 2y = 7
∴ -23y = 7 – 30
∴ -23y = -23
∴ y = \(\frac { -23 }{ -23 } \)
Substituting y = 1 in equation (i), we get
x = 10 – 7 (1)
= 10 – 7 = 3
∴ (x, y) = (3, 1) is the solution of the given simultaneous equations.

iii. 2x – 3y = 9 …(i)
2x + y = 13 …(ii)
Subtracting equation (ii) from (i), we get
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.1 3
∴ (x, y) = (6, 1) is the solution of the given simultaneous equations.

iv. 5m – 3n = 19 …(i)
m – 6n = -7
∴ m = 6n – 7 …(ii)
Substituting m = 6n – 7 in equation (i), we get
5(6n – 7) – 3n = 19
∴ 30n – 35 – 3n = 19
∴ 27n = 19 + 35
∴ 27n = 54
∴ n = \(\frac { 54 }{ 27 } \) = 2
Substituting n = 2 in equation (ii), we get
m = 6(2) – 7
= 12 – 7 = 5
∴ (m, n) = (5, 2) is the solution of the given simultaneous equations.

v. 5x + 2y = -3 …(i)
x + 5y = 4
∴ x = 4 – 5y …(ii)
Substituting x = 4 – 5y in equation (i), we get
5(4 – 5y) + 2y = -3
∴ 20 – 25y + 2y = -3
∴ -23y = -3 – 20
∴ -23y = -23
∴ y = \(\frac { -23 }{ -23 } \) = 1
Substituting y = 1 in equation (ii), we get
x = 4 – 5(1)
= 4 – 5 = -1
∴ (x, y) = (-1, 1) is the solution of the given simultaneous equations.

Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.1 4
Substituting y = 3 in equation (i), we get
x = 10 – 3(3)
= 10 – 9 = 1
∴ (x, y) = (1, 3) is the solution of the given simultaneous equations.

vii. 99x + 101 y = 499 …(i)
101 x + 99y = 501 …(ii)
Adding equations (i) and (ii), we get
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.1 5
Substituting x = 3 in equation (iii), we get
3 + y = 5
∴ y = 5 – 3 = 2
∴ (x, y) = (3, 2) is the solution of the given simultaneous equations.

viii. 49x – 57y = 172 …(i)
57x – 49y = 252 …(ii)
Adding equations (i) and (ii), we get
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.1 6
Substituting x = 7 in equation (iv), we get
7 + y = 10
∴ y = 10 – 7 = 3
∴ (x, y) = (7, 3) is the solution of the given simultaneous equations.

Complete the following table. (Textbook pg. no. 1)
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.1 7

Question 1.
Solve: 3x+ 2y = 29; 5x – y = 18 (Textbook pg. no. 3)
Solution:
3x + 2y = 29 …(i)
and 5x- y = 18 …(ii)
Let’s solve the equations by eliminating ‘y’.
Fill suitably the boxes below.
Multiplying equation (ii) by 2, we get
Maharashtra Board Class 10 Maths Solutions Chapter 1 Linear Equations in Two Variables Ex 1.1 8

Maharashtra Board Class 10 Maths Solutions