### NCERT Solutions for Class 10 Maths Exercise 3.2

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1. Form the pair of linear equations in the following problems, and find their solutions graphically.
(i). 10 students of class X took part in a mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

**Ans. (i)** Let number of boys who took part in the quiz = x
Let number of girls who took part in the quiz = y
According to given conditions, we have
*x *+ *y *= 10… (1)
And, *y *= *x *+ 4
⇒ *x *– *y *= −4 … (2)
For equation x + y = 10, we have following points which lie on the line
For equation x – y = –4, we have following points which lie on the line
We plot the points for both of the equations to find the solution.
We can clearly see that the intersection point of two lines is (3, 7).
Therefore, number of boys who took park in the quiz = 3 and, number of girls who took part in the quiz = 7.

(ii) 5 pencils and 7 pens together cost Rs 50, whereas 7 pencils and 5 pens together cost Rs. 46. Find the cost of one pencil and that of one pen.
Ans. (ii) Let cost of one pencil = Rs x and Let cost of one pen = Rs y

According to given conditions, we have
5*x *+ 7*y *= 50… (1)
7*x *+ 5*y *= 46… (2)
For equation 5x + 7y = 50, we have following points which lie on the line
For equation 7x + 5y = 46, we have following points which lie on the line
We can clearly see that the intersection point of two lines is (3, 5).
Therefore, cost of pencil = Rs 3 and, cost of pen = Rs 5

**Ans. (i)**Let number of boys who took part in the quiz = x

*x*+

*y*= 10… (1)

*y*=

*x*+ 4

*x*–

*y*= −4 … (2)

Therefore, number of boys who took park in the quiz = 3 and, number of girls who took part in the quiz = 7.

*x*+ 7

*y*= 50… (1)

*x*+ 5

*y*= 46… (2)

###### 2. On comparing the ratios , find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:

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(i) 5*x *− 4*y *+ 8 = 0

7*x *+ 6*y *– 9 = 0
Ans. (i) 5*x *− 4*y *+ 8 = 0, 7*x *+ 6*y *– 9 = 0
Comparing equation 5*x *− 4*y *+ 8 = 0 with and 7*x *+ 6*y *– 9 = 0 with
We get, , , , , ,
We have because
Hence, lines have unique solution which means they intersect at one point.

(ii) 9*x *+ 3*y *+ 12 = 0

18*x *+ 6*y *+ 24 = 0
Ans. 9*x *+ 3*y *+ 12 = 0, 18*x *+ 6*y *+ 24 = 0
Comparing equation 9*x *+ 3*y *+ 12 = 0 with and 18*x *+ 6*y *+ 24 = 0 with ,
We get, , , , , ,
We have because
Hence, lines are coincident.

(iii) 6*x *− 3*y *+ 10 = 0
2*x *– *y *+ 9 = 0
**Ans. ** 6*x *− 3*y *+ 10 = 0, 2*x *– *y *+ 9 = 0
Comparing equation 6*x *− 3*y *+ 10 = 0 with and 2*x *– *y *+ 9 = 0 with ,
We get, , , , , ,
We have because
Hence, lines are parallel to each other.

*x*− 4

*y*+ 8 = 0

7

*x*+ 6

*y*– 9 = 0

*x*− 4

*y*+ 8 = 0, 7

*x*+ 6

*y*– 9 = 0

*x*− 4

*y*+ 8 = 0 with and 7

*x*+ 6

*y*– 9 = 0 with

(ii) 9

*x*+ 3*y*+ 12 = 0
18

Ans. 9*x*+ 6*y*+ 24 = 0*x*+ 3

*y*+ 12 = 0, 18

*x*+ 6

*y*+ 24 = 0

*x*+ 3

*y*+ 12 = 0 with and 18

*x*+ 6

*y*+ 24 = 0 with ,

Hence, lines are coincident.

(iii) 6

*x*− 3*y*+ 10 = 0
2

*x*–*y*+ 9 = 0**Ans.**6

*x*− 3

*y*+ 10 = 0, 2

*x*–

*y*+ 9 = 0

*x*− 3

*y*+ 10 = 0 with and 2

*x*–

*y*+ 9 = 0 with ,

###### 3. On comparing the ratios , find out whether the following pair of linear equations are consistent, or inconsistent.

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(i) 3*x *+ 2*y *= 5, 2*x *− 3*y *= 8
Ans. (i) 3*x *+ 2*y *= 5, 2*x *− 3*y *= 7
Comparing equation 3*x *+ 2*y *= 5 with and 2*x *− 3*y *= 7 with ,
We get,
and
Here which means equations have unique solution.
Hence they are consistent.

**(ii)** 2*x *− 3*y *= 7, 4*x *− 6*y *= 9
Ans.(ii) 2*x *− 3*y *= 8, 4*x *− 6*y *= 9
Comparing equation 2*x *− 3*y *= 8 with and 4*x *− 6*y *= 9 with ,
We get,
Here because
Therefore, equations have no solution because they are parallel.
Hence, they are inconsistent.

(iii) 9*x *− 10*y *= 14
Ans.(iii) 9*x *− 10*y *= 14
Comparing equation with and 9*x *− 10*y *= 14 with ,
We get, , ,
and
Here
Therefore, equations have unique solution.
Hence, they are consistent.

**(iv)** 5*x *− 3*y *= 11, −10*x *+ 6*y *= −22
Ans.(iv) 5*x *− 3*y *= 11, −10*x *+ 6*y *= −22
Comparing equation 5*x *− 3*y *= 11 with and −10*x *+ 6*y *= −22 with ,
We get,
and
Here
Therefore, the lines have infinite many solutions.
Hence, they are consistent.

*x*+ 2

*y*= 5, 2

*x*− 3

*y*= 8

*x*+ 2

*y*= 5, 2

*x*− 3

*y*= 7

*x*+ 2

*y*= 5 with and 2

*x*− 3

*y*= 7 with ,

Hence they are consistent.

**(ii)**2

*x*− 3

*y*= 7, 4

*x*− 6

*y*= 9

*x*− 3

*y*= 8, 4

*x*− 6

*y*= 9

*x*− 3

*y*= 8 with and 4

*x*− 6

*y*= 9 with ,

Hence, they are inconsistent.

*x*− 10

*y*= 14

*x*− 10

*y*= 14

*x*− 10

*y*= 14 with ,

Hence, they are consistent.

**(iv)**5

*x*− 3

*y*= 11, −10

*x*+ 6

*y*= −22

*x*− 3

*y*= 11, −10

*x*+ 6

*y*= −22

*x*− 3

*y*= 11 with and −10

*x*+ 6

*y*= −22 with ,

###### 4. Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:

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(i) *x *+ *y *= 5, 2*x *+ 2*y *= 10
Ans. (i) *x *+ *y *= 5, 2*x *+ 2*y *= 10
For equation x + y – 5 = 0, we have following points which lie on the line
For equation 2x + 2y – 10 = 0, we have following points which lie on the line
We can see that both of the lines coincide. Hence, there are infinite many solutions. Any point which lies on one line also lies on the other. Hence, by using equation (*x *+ *y *– 5 = 0), we can say that *x *= 5 − *y*
We can assume any random values for y and can find the corresponding value of x using the above equation. All such points will lie on both lines and there will be infinite number of such points.

(ii) *x *– *y *= 8, 3*x *− 3*y *= 16
Ans.(ii) *x *– *y *= 8, 3*x *− 3*y *= 16
For x – y = 8, the coordinates are:
And for 3x – 3y = 16, the coordinates
Plotting these points on the graph, it is clear that both lines are parallel. So the two lines have no common point. Hence the given equations have no solution and lines are inconsistent.

(iii) 2*x *+ *y *= 6, 4*x *− 2*y *= 4
Ans. (iii) 2*x *+ *y *= 6, 4*x *− 2*y *= 4
For equation 2x + y – 6 = 0, we have following points which lie on the line
For equation 4x – 2y – 4 = 0, we have following points which lie on the line
We can clearly see that lines are intersecting at (2, 2) which is the solution.
Hence x = 2 and y = 2 and lines are consistent.

(iv) 2*x *− 2*y *– 2 = 0, 4*x *− 4*y *– 5 = 0

Ans.(iv) 2*x *− 2*y *– 2 = 0, 4*x *− 4*y *– 5 = 0
For 2x – 2y – 2 = 0, the coordinates are:
And for 4x – 4y – 5 = 0, the coordinates
Plotting these points on the graph, it is clear that both lines are parallel. So the two lines have no common point. Hence the given equations have no solution and lines are inconsistent.

*x*+

*y*= 5, 2

*x*+ 2

*y*= 10

*x*+

*y*= 5, 2

*x*+ 2

*y*= 10

*x*+

*y*– 5 = 0), we can say that

*x*= 5 −

*y*

(ii)

*x*–

*y*= 8, 3

*x*− 3

*y*= 16

*x*–

*y*= 8, 3

*x*− 3

*y*= 16

(iii) 2

*x*+*y*= 6, 4*x*− 2*y*= 4*x*+

*y*= 6, 4

*x*− 2

*y*= 4

(iv) 2

*x*− 2

*y*– 2 = 0, 4

*x*− 4

*y*– 5 = 0

Ans.(iv) 2

*x*− 2

*y*– 2 = 0, 4

*x*− 4

*y*– 5 = 0

###### 5. Half the perimeter of a rectangle garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.

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Ans. Let length of rectangular garden = x metres
Let width of rectangular garden = y metres
According to given conditions, perimeter = 36 m
x + y = 36 ……(i)
And x = y + 4
⇒ *x *– *y *= 4……..(ii)
Adding eq. (i) and (ii),
2x = 40
x = 20 m
Subtracting eq. (ii) from eq. (i),
2y = 32
y = 16 m
Hence, length = 20 m and width = 16 m

*x*–

*y*= 4……..(ii)

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