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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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