R.D. Sharma Solutions Class 7 | Math Chapter 8 Linear Equations In One Variable Exercise 8.1

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Chapter 8: Linear Equations in One Variable Exercise – 8.1

Question: 1

Verify by substitution that:

(i) x = 4 is the root of 3x – 5 = 7

(ii) x = 3 is the root of 5 + 3x = 14

(iii) x = 2 is the root of 3x – 2 = 8x – 12

(iv) x = 4 is the root of 3x/2 = 6

(v) y = 2 is the root of y – 3 = 2y – 5

(vi) x = 8 is the root of (1/2)x + 7 = 11

Solution:

(i) x = 4 is the root of 3x - 5 = 7.

Now, substituting x = 4 in place of ‘x’ in the given equation 3x - 5 = 7,

3(4) – 5 = 7

12 – 5 = 7

7 = 7

Since, LHS = RHS

Hence, x = 4 is the root of 3x - 5 = 7.

(ii). x = 3 is the root of 5 + 3x = 14.

Now, substituting x = 3 in place of ‘x’ in the given equation 5 + 3x = 14,

5 + 3(3) = 14

5 + 9 = 14

14 = 14

Since, LHS = RHS

Hence, x = 3 is the root of 5 + 3x = 14.

(iii). x = 2 is the root of 3x – 2 = 8x – 12.

Now, substituting x = 2 in place of ‘x’ in the given equation 3x –2 = 8x – 12,

3(2) – 2 = 8(2) – 12

6 – 2 = 16 – 12

4 = 4

Since, LHS = RHS

Hence, x = 2 is the root of 3x – 2 = 8x – 12.

(iv) x = 4 is the root of 3x/2 = 6.

Now, substituting x = 4 in place of ‘x’ in the given equation 3x/2 = 6,

(3× 4)/2 = 6

12/2 = 6

6 = 6

Since, LHS = RHS

Hence, x = 4 is the root of 3x/2 = 6.

(v). y = 2 is the root of y – 3 = 2y – 5.

Now, substituting y = 2 in place of ‘y’ in the given equation y – 3 = 2y – 5,

2 – 3 = 2(2) – 5

-1 = 4 – 5

-1 = -1

Since, LHS = RHS

Hence, y = 2 is the root of y – 3 = 2y – 5.

(vi). x = 8 is the root of 12x + 7 = 11.

Now, substituting x = 8 in place of ‘x’ in the given equation 12x + 7 = 11,

12(8) + 7 =11

4 + 7 = 11

11 = 11

Since, LHS = RHS

Hence, x = 8 is the root of 12x + 7 = 11.

 

Question: 2

Solve each of the following equations by trial and error method:

(i) x + 3 = 12

(ii) x – 7 = 10

(iii) 4x = 28

(iv) x/2 + 7 = 11

(v) 2x + 4 = 3x

(vi) x/4 = 12

(vii) 15/x = 3

(viii) x/18 = 20

Solution:

(i) x + 3 = 12

Here, LHS = x + 3 and RHS = 12


	
		
	
	
		
			x
			LHS
			RHS
			Is LHS = RHS
		
		
			1
			1 + 3 = 4
			12
			No
		
		
			2
			2 + 3 = 5
			12
			No
		
		
			3
			3 + 3 = 6
			12
			No
		
		
			4
			4 + 3 = 7
			12
			No
		
		
			5
			5 + 3 = 8
			12
			No
		
		
			6
			6 + 3 = 9
			12
			No
		
		
			7
			7 + 3 = 10
			12
			No
		
		
			8
			8 + 3 = 11
			12
			No
		
		
			9
			9 + 3 = 12
			12
			Yes
		
	




Therefore, if x = 9, LHS = RHS.

Hence, x = 9 is the solution to this equation.

(ii) x – 7 = 10

Here, LHS = x – 7 and RHS = 10.


	
		
	
	
		
			x
			LHS
			RHS
			Is LHS = RHS
		
		
			9
			9 – 7 = 2
			10
			No
		
		
			10
			10 – 7 = 3
			10
			No
		
		
			11
			11 – 7 = 4
			10
			No
		
		
			12
			12 – 7 = 5
			10
			No
		
		
			13
			13 – 7 = 6
			10
			No
		
		
			14
			14 – 7 = 7
			10
			No
		
		
			15
			15 – 7 = 8
			10
			No
		
		
			16
			16 – 7 = 9
			10
			No
		
		
			17
			17 – 7 = 10
			10
			Yes
		
	




Therefore, if x = 17, LHS = RHS.

Hence, x = 17 is the solution to this equation.

(iii) 4x = 28

Here, LHS = 4x and RHS = 28.




	
		
	
	
		
			x
			LHS
			RHS
			Is LHS = RHS
		
		
			1
			4 x 1 = 4
			28
			No
		
		
			2
			4 x 2 = 8
			28
			No
		
		
			3
			4 x 3 = 12
			28
			No
		
		
			4
			4 x 4 = 16
			28
			No
		
		
			5
			4 x 5 = 20
			28
			No
		
		
			6
			4 x 6 = 24
			28
			No
		
		
			7
			4 x 7 = 28
			28
			Yes
		
	


Therefore, if x = 7, LHS = RHS

Hence, x = 7 is the solution to this equation.

(iv) x/2 + 7 = 11

Here, LHS = x/2 + 7 and RHS = 11.

Since RHS is a natural number, x/2 must also be a natural number, so we must substitute values of x that are multiples of 2.


	
		
	
	
		
			x
			LHS
			RHS
			Is LHS = RHS
		
		
			2
			2/2 + 7 = 8
			11
			No
		
		
			4
			4/2 + 7 = 9
			11
			No
		
		
			6
			6/2 + 7 = 10
			11
			No
		
		
			8
			8/2 + 7 = 11
			11
			Yes
		
	


Therefore, if x = 8, LHS = RHS.

Hence, x = 8 is the solution to this equation.

(v) 2x + 4 = 3x

Here, LHS = 2x + 4 and RHS = 3x.


	
		
	
	
		
			x
			LHS
			RHS
			Is LHS = RHS
		
		
			1
			2(1) + 4 = 6
			3(1) = 3
			No
		
		
			2
			2(2) + 4 = 8
			3(2) = 6
			No
		
		
			3
			2(3) + 4 =10
			3(3) = 9
			No
		
		
			4
			2(4) + 5 = 12
			 3(4) = 12
			Yes
		
	




Therefore, if x = 4, LHS = RHS.

Hence, x = 4 is the solution to this equation.

(vi) x/4 = 12

Here, LHS = x/4 and RHS = 12.

Since RHS is a natural number, x/4 must also be a natural number, so we must substitute values of x that are multiples of 4.


	
		
	
	
		
			X
			LHS
			RHS
			Is LHS = RHS
		
		
			16
			16/4 = 4
			12
			No
		
		
			20
			20/4 = 5
			12
			No
		
		
			24
			24/4 = 6
			12
			No
		
		
			28
			28/4 = 7
			12
			No
		
		
			32
			32/4 = 8
			12
			No
		
		
			36
			36/4 = 9
			12
			No
		
		
			40
			40/4 = 10
			12
			No
		
		
			44
			44/4 = 11
			12
			No
		
		
			48
			48/4 = 12
			12
			Yes
		
	


Therefore, if x = 48, LHS = RHS.

Hence, x = 48 is the solution to this equation.

(vii) 15x = 3

Here, LHS = 15x and RHS = 3.

Since RHS is a natural number, 15x must also be a natural number, so we must substitute values of x that are factors of 15.


	
		
	
	
		
			x
			LHS
			RHS
			Is LHS = RHS
		
		
			1
			15/1= 15
			3
			No
		
		
			3
			15/3 = 5
			3
			No
		
		
			5
			15/5 = 3
			3
			Yes
		
	


Therefore, if x = 5, LHS = RHS.

Hence, x = 5 is the solution to this equation.

(viii) x/18 = 20

Here, LHS = x/18 and RHS = 20.

Since RHS is a natural number, x/18 must also be a natural number, so we must substitute values of x that are multiples of 18.


	
		
	
	
		
			X
			LHS
			RHS
			Is LHS = RHS
		
		
			324
			324/18 = 18
			20
			No
		
		
			342
			342/18 = 19
			20
			No
		
		
			360
			360/18 = 20
			20
			Yes
		
	


Therefore, if x = 360, LHS = RHS.

Hence, x = 360 is the solution to this equation.

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Chapter 8: Linear Equations in One Variable Exercise – 8.1

Question: 1

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Question: 2

Solution:

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x	LHS	RHS	Is LHS = RHS
1	1 + 3 = 4	12	No
2	2 + 3 = 5	12	No
3	3 + 3 = 6	12	No
4	4 + 3 = 7	12	No
5	5 + 3 = 8	12	No
6	6 + 3 = 9	12	No
7	7 + 3 = 10	12	No
8	8 + 3 = 11	12	No
9	9 + 3 = 12	12	Yes

x	LHS	RHS	Is LHS = RHS
1	4 x 1 = 4	28	No
2	4 x 2 = 8	28	No
3	4 x 3 = 12	28	No
4	4 x 4 = 16	28	No
5	4 x 5 = 20	28	No
6	4 x 6 = 24	28	No
7	4 x 7 = 28	28	Yes