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### Solved Examples

**Question 1:**The perimeter of a rectangular field is 81 meter. If the length of the field is 3 meter, find the breadth of the field.

**Solution:**

Perimeter of the rectangular field = 81

Length of the field = 3

Let breadth of the field = x

[Perimeter of the rectangle = 2(Length + Breadth)]

=> 81 = 2(3 + x)

=> 81 = 6 + 2x

Subtract 6 from both sides

=> 81 - 6 = 6 + 2x - 6

=> 75 = 2x

Divide each side by 2

=> $\frac{75}{2}$ = x

Hence, breadth of the rectangular field = $\frac{75}{2}$ meter

Length of the field = 3

Let breadth of the field = x

[Perimeter of the rectangle = 2(Length + Breadth)]

=> 81 = 2(3 + x)

=> 81 = 6 + 2x

Subtract 6 from both sides

=> 81 - 6 = 6 + 2x - 6

=> 75 = 2x

Divide each side by 2

=> $\frac{75}{2}$ = x

Hence, breadth of the rectangular field = $\frac{75}{2}$ meter

**Question 2:**Factor 6x

^{2}- 7x + 2

**Solution:**

6x

= 2x(3x - 2) - (3x - 2)

= (2x - 1)(3x - 2)

=> 6x

^{2}- 7x + 2 = 6x^{2}- 4x - 3x + 2= 2x(3x - 2) - (3x - 2)

= (2x - 1)(3x - 2)

=> 6x

^{2}- 7x + 2 = (2x - 1)(3x - 2)**Question 3:**The sum of the digits of a two-digit number is 10. If 54 is added to this number, then the digits are reversed. Find the number.

**Solution:**

Let the ten's place and the unit place digits of the number be x and y respectively.

=> The number is 10x + y

If the digits are reversed, the new number becomes 10y + x

Step 1:

The statement states:

x + y = 10 ..............................(1)

and (10x + y) + 54 = 10y + x ..............................(2)

Step 2:

(1) => y = 10 - x ...........................(3)

(2) => 10x - x + y - 10y + 54 = 0

=> 9x - 9y + 54 = 0 ............................(4)

Put equation (3) in (4)

=> 9x - 9(10 - x) + 54 = 0

=> 9x - 90 + 9x + 54 = 0

=> 18x - 36 = 0

=> 18x = 36

Divide each side by 18

=> x = 2

Step 3:

Put x = 2 in equation (1)

=> 2 + y = 10

=> y = 8

Hence the number is 28. answer

=> The number is 10x + y

If the digits are reversed, the new number becomes 10y + x

Step 1:

The statement states:

x + y = 10 ..............................(1)

and (10x + y) + 54 = 10y + x ..............................(2)

Step 2:

(1) => y = 10 - x ...........................(3)

(2) => 10x - x + y - 10y + 54 = 0

=> 9x - 9y + 54 = 0 ............................(4)

Put equation (3) in (4)

=> 9x - 9(10 - x) + 54 = 0

=> 9x - 90 + 9x + 54 = 0

=> 18x - 36 = 0

=> 18x = 36

Divide each side by 18

=> x = 2

Step 3:

Put x = 2 in equation (1)

=> 2 + y = 10

=> y = 8

Hence the number is 28. answer