Question

Determinati numerele x y z stiind ca:


a) x/6=y/7=z/8                                    
x+y+z=20                                                             

    b) x/5=y/3=z/7 
 x+2y-z=192    

c) x/14=y/11=z/12
x+y-z=39

Answer 1

a).  x/6=y/7=z/8=(x+y+z)/(6+7+8)=20/21
         ⇒ x = 6 * 20/21 = 2*20 / 7 = 40/7 ;
              y = 7 * 20/21 =1*20 / 3 = 20/3  ;
              z = 4 * 20/21 =               80/21  ;
b).  x/5=y/3=z/7  ⇔ x/5=2y/6=z/7=(x+2y-z) / (5+6-7)=192 / 4 = 48 ;
         ⇒  x=5 * 48 =240  ;   2y= 6 * 48=288  deci  y= 144 ;  z = 7 * 48 =336
c).   x/14=y/11=z/12 =(x+y-z)/(14+11-12)=39/13 = 3
         ⇒  x = 14 * 3 =42   ;   y = 11 * 3 = 33   și   z = 12 * 3 =36  ;

Answer 2

a. 7x=6y => y= 7x/6 ; 8x=6z =.> z= 8x/6
x+ 7x/6 + 8x/6 = 20
6x+7x+8x = 120
21x = 120 => x = 120/21 = 40/7
y= 40/6
z= 160/21

b. 7x= 5z => z= 7x/5; 3x=5y => y = 3x/5
x+ 6x/5-7x/5 = 192
5x+6x-7x =  960 => 4x=960 => x= 240
y= 144; z = 336

c. 12x=14z => z=12x/14 ; 11x=14y => y= 11x/14
x+ 11x/14 - 12x/14 = 39
14x+11x-12x = 546
13x=546 => x = 42 ; y= 33 ; z = 36