Solve the following equations and verify the solution: 3 4 y + 2 y = 1 2 + 4 y - 3

Given linear equation is: 3 4 y + 2 y = 1 2 + 4 y - 3 By simplifying the equation, we get, 11 y 4 = 4 y - 5 2 Subtracting 4 y on both sides, we get 11 y 4 - 4 y = - 5 2 11 y - 16 y 4 = - 5 2 - 5 y 4 = - 5 2 Dividing both sides by - 5 4 , we get y = - 5 2 × - 4 5 = 2 Hence, y = 2 is the solution to equation 3 4 y + 2 y = 1 2 + 4 y - 3 Verification: Substituting y = 2 in left-hand side of the equation LHS ⇒ 3 4 y + 2 y = 3 4 × 2 + 2 × 2 = 3 2 + 4 = 11 2 Substituting y = 2 in right-hand side of the equation RHS ⇒ 1 2 + 4 y - 3 = 1 2 + 4 × 2 - 3 = 1 2 + 5 = 11 2 Since LHS = RHS Hence, verified.

Mr Khan won a cash prize of a certain amount in the jackpot of a carnival. He gave half of this money and an additional amount of ₹ 10 , 000 to his wife. He also gave 1 3 of the money received and additional amount of ₹ 3000 to his son. If his wife got double the amount his son got, find the amount of money Mr Khan won in the jackpot.

Given: Mr Khan won a cash prize of a certain amount in the jackpot of a carnival. He gave half of this money and an additional amount of ₹ 10 , 000 to his wife. He also gave 1 3 of the money received and additional amount of ₹ 3000 to his son. Let amount of prize money be Rs x . It is given that, Khan gave half of this money and an additional amount of ₹ 10 , 000 to his wife. He also gave 1 3 of the money received and additional amount of ₹ 3000 to his son. So, 1 2 x + 10000 = 2 1 3 x + 3000 ⇒ x 2 + 10000 = 2 x 3 + 6000 ⇒ x 2 - 2 x 3 = 6000 - 10000 [By transposing] ⇒ 3 x - 4 x 6 = - 4000 ⇒ - x = - 24000 ⇒ x = 24000 Therefore, the amount of money Mr Khan won in the jackpot is Rs 24000 .