Răspuns:
Explicație pas cu pas:
[tex]\frac{a}{3}= \frac{b}{4}= \frac{c}{6}= k,~coeficient~de~proportionalitate\\\\deci~a=3k,~b=4k,~c=6k,~inlocuim~in~2a+5b+3c=88,~obtinem,~2*3k+5*4k+3*6k=88,~6k+20k+18k=88,~44k=88,~k=2.\\Deci~a=3*2=6,~b=4*2=8,~c=6*2=12.\\Atunci~4a+3b-4c=4*6+3*4-4*12=24+12-48=36-48=-12.\\\frac{1}{a^{2}}+ \frac{1}{b^{2}}+\frac{1}{c^{2}}=\frac{1}{36} +\frac{1}{64}+\frac{1}{144}=\frac{16+9+4}{576}=\frac{29}{576}[/tex]
Am aflat cmmmc [36,64,144]=576, deoarece
36=2²·3², 64=2⁶, 144=2⁴·3², deci [36,64,144]=2⁶·3²=576
Răspuns:
Explicație pas cu pas:
a/3 = b/4 = c/6 = k → coeficient de proportionalitate
⇒ a = 3 k; b = 4 k si c = 6 k
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2 a + 5 b + 3 c = 88
2 × 3 k + 5 × 4 k + 3 × 6 k = 88
6 k + 20 k + 18 k = 88
44 k = 88
k = 88 : 44 ⇒ k = 2
__________________
a = 3 × 2 ⇒ a = 6
b = 4 × 2 ⇒ b = 8
c = 6 × 2 ⇒ c = 12
________________
4 a + 3 b - 4 c = 0
= 4 × 6 + 3 × 8 - 4 × 12 =
= 24 + 24 - 48 =
= 48 - 48 =
= 0
________________
1/a² + 1/b² + 1/c² = 29/576
= 1/6² + 1/8² + 1/12² =
= 1/36 + 1/64 + 1/144 =
→ aflam cel mai mic multiplu comun al numitorilor 36, 64 si 144:
36 = 4 × 9
64 = 4³
144 = 4 × 36 = 4² × 9
_________________
c.m.m.m.c al numitorilor: 4³ × 9 = 144 × 4 = 576
→ aduc fractile la acelasi numitor 144, amplificand prima fractie cu 16, a doua fractie cu 9
= 16/576 + 9/576 + 4/576 =
= 29/576
