Question

Determinati numerele x si y stiind ca : a)sunt invers proportionale cu 2 si 3 iar x-y=10 b)sunt invers proportionale cu 3 si 4 iar x-y=63 c)sunt invers proportionale cu 2 si 3 iar x la puterea a 2 + y la puterea a 2 a =208 d)sunt invers proportionale cu 3 si 4 iar x la puterea a 2 a -y la puterea a 2a =175 e)sunt invers proportionale cu 3 si 5 iar xy =135

Answer 1

a)x-y=10 {x,y}ip{2,3}⇒2x=3y=k 2x=k⇒x=k/2 3y=k⇒y=k/3 k/2-k/3=10⇒ (3k-2k)/6=10 k/6=10⇒ k=6·10⇒ k=60 x=60/2=30 y=60/3=20 b) x-y=63 3x=4y=k⇒ 3x=k⇒x=k/3 si 4y=k⇒y=k/4 k/3-k/4=63⇒ (4k-3k)/12=63 k=63·12=756 x=756/3⇒x=253 y=756/4⇒y=189 c)x²+y²=208 2x=3y=k x=k/2 si y=k/3 k²/4+k²/9=208 (9k²+4k²)/36=208 13k²/36=208 k²=208·36/13 k²=576 k=√576⇒k=24 x=24/2⇒x=12 y=24/3⇒y=8 d) x²+y²=175 3x=4y=k 3x=k⇒x=k/3 4y=k⇒y=k/4 k²/9+k²/16=175 (16k²+9k²)144=175 25k²/144=175 k²=175·144/25 k²=1008 k=√1008⇒k=31,75 x=31,75/3⇒x=10,58 y=31,75/4⇒y=7,93 e) xy=135 3x=5y=k 3x=k⇒x=k/3 5y=k⇒y=k/5 k/3·k/5=135 k²/15=135 k²=135·15 k²=2025 k=√2025⇒k=45 x=45/3⇒x=15 y=45/5⇒y=9