Linear Equations in One Variable - 8th math

Solve the following equations

Question (5)

Solution

Given,

After cross multiplication, we get

3(7y + 4) = –4(y + 2)

⇒ 21y + 12 = –4y – 8

After transposing –4y to LHS, we get

⇒ 21y + 12 + 4y = –8

After transposing 12 to RHS, we get

⇒ 21y + 4y = –8 – 12

⇒ 25y = –20

After transposing 25 to RHS, we get

y = –20/25

⇒ y = –4/5 Answer

Checking of Result

Given,

After substituting the value of y =–4/5 in LHS, we get

= –4/3 = RHS Proved

Question (6) The ages of Hari and Harry are in the ratio 5:7. Four years from now the ratio of their ages will be 3:4. Find their present ages.

Solution:

Given, Present ages of Hari and Harry = 5:7

And after 4 years from now the ratio of their ages = 3:4

Thus, their present ages = ?

Let the present age of Hari = 5x

And the Present age of Harry = 7x

After four years from now

The age of Hari would be = 5x + 4

And the age of Harry will be = 7x + 4

Now, according to question, the ratio of their ages after 4 years = 3:4

Thus,

After cross multiplication, we get

4(5x + 4) = 3(7x + 4)

⇒ 20x + 16 = 21x + 12

After transposing 20x to RHS, we get

⇒ 16 = 21x + 12 – 20x

After transposing 12 to LHS, we get

⇒16 – 12 = 21x – 20x

⇒ 4 = x

⇒ x = 4

Thus, present age of Hari = 5x

= 5 × 4 = 20

Thus, present age of Hari = 20

And similarly, the present age of Harry = 7x

= 7 × 4 = 28

Thus, present age of Harry = 28

Thus, present age of Hari = 20 and the present age of Harry = 28 Answer

Checking of Result

Given, Present ages of Hari and Harry = 5:7

And after 4 years from now the ratio of their ages = 3:4

Thus, their present ages = ?

We have, present age of Hari = 20 and the present age of Harry = 28

Thus, after 4 years now,

The age of Hari after 4 years now = Present age of Hari + 4

= 20 + 4 = 24

Thus, age of Hari after 4 years now = 24 years

Similarly, the age of Harry after 4 years now = the present age of Harry + 4

= 28 + 4 = 32 years

Thus, age of Harry after 4 years now = 28 years

Thus, ratio of present age of Hari and Harry = 20 : 28

= 5 : 7

And the ratio of age of Hari and Harry after 4 years now = 24:32

= 3 : 4

Thus, ratio of present age of Hari and Harry = 5:7 and ratio of their ages after 4 years now = 3:4 Proved

Question (7) The denominator of a rational number is greater than its numerator by 8. If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is 3/2. Find the rational number.

Solution

Given, the denominator of a rational number = numerator + 8

And, the rational number = when numerator +17 and denominator – 1 = 3/2

Thus, rational number =?

Let the numerator of the given rational number = n

Thus, as per question, denominator = n + 8

Thus, rational number = n/(n+8)

Now, According to question,

numerator +17 and denominator – 1 = 3/2

Thus,

After cross multiplication, we get

3(n + 7) = 2(n + 17)

⇒ 3n + 21 = 2n + 34

After transposing 2n to LHS, we get

⇒ 3n + 21 – 2n = 34

After transposing 21 to RHS, we get

⇒ 3n – 2n = 34 – 21

⇒ n = 13

Now, since denominator = n + 8

Thus, after substituting the value of n = 13, we get

The denominator = 13 + 8 = 21

Thus, nominator = 13 and denominator = 21

Thus, required rational number = 13/21 Answer

Checking of Result

Given, the denominator of a rational number = numerator + 8

And, the rational number = when numerator +17 and denominator – 1 = 3/2

Thus, rational number =?

We have rational number =

Now, after increasing numerator by 17 and decreasing denominator by 1, we get

Thus, after increasing numerator by 17 and decreasing denominator by 1 rational number becomes equal to Proved