[tex]\displaystyle\\x;~y;~z~~\text{sunt invers proportionale cu}~~4;~10;~5\\\\4x=10y=5z=k\\\\x=\frac{k}{4}\\\\y=\frac{k}{10}\\\\z=\frac{k}{5}\\\\x+3z=34\\\\\frac{k}{4}+3\cdot\frac{k}{5}=34\\\\\frac{k\cdot5}{4\cdot5}+\frac{3\cdot k\cdot4}{5\cdot4}=34\\\\\frac{5k}{20}+\frac{12k}{20}=34\\\\\frac{5k+12k}{20}=34\\\\\frac{17k}{20}=34\\\\17k=34\cdot20\\\\k=\frac{34\cdot20}{17}\\\\k=2\cdot20\\\\\boxed{k=40}[/tex]
[tex]\displaystyle\\\\x=\frac{k}{4}=\frac{40}{4}=\boxed{10}\\\\y=\frac{k}{10}=\frac{40}{10}=\boxed{4}\\\\z=\frac{k}{5}=\frac{40}{5}=\boxed{8}[/tex]
Răspuns
Explicație pas cu pas:
[ x; y; z ] i. p. [ 4; 10; 5 ]
x / ( 1 / 4 ) = y / ( 1 / 10 ) = z / ( 1 / 5 ) = k
x / ( 1 / 4 ) = x : 1 / 4 = x · 4 / 1 = 4 x
⇒ 4 x = 10 y = 5 z = k ⇒ x = k / 4; y = k / 10 si z = k / 5
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x + 3 z = 34
k / 4 + 3 × k / 5 = 34
→ c.m.m.m.c al numitorilor 4 si 5 este 20
= 5 k + 4 × 3 k = 20 × 34
5 k + 12 k = 680
17 k = 680
k = 680 : 17 ⇒ k = 40
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x = k / 4 ⇒ x = 40 / 4 ⇒ x = 10
y = k / 10 ⇒ y = 40 / 10 ⇒ y = 4
z = k / 5 ⇒ z = 40 / 5 ⇒ z = 8