Question

a) 3x=2y, 7y=5z, x-y+z=32

b) x-1/2= y-2/3=z-3/4, 2x+3y-z=50

Answer 1

a)

3x=2y => 21x=14y

7y=5x => 14y=10z

=> 21x=14y=10z

\(\Rightarrow\frac{x}{\frac{1}{21}}=\frac{y}{\frac{1}{14}}=\frac{z}{\frac{1}{10}}=\frac{x-y+z}{\frac{1}{21}-\frac{1}{14}+\frac{1}{10}}=\frac{32}{\frac{10-15+21}{210}}=32\cdot\frac{210}{16}=420\)

\(x=\frac{1}{21}\cdot420=20\)

\(y=\frac{1}{14}\cdot420=30\)

\(z=\frac{1}{10}\cdot420=42\).

b)

\(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}=\frac{2x-2}{4}=\frac{3y-6}{9}=\frac{-z+3}{-4}=\frac{2x-2+3y-6-z+3}{4+9-4}=\frac{45}{9}=5\)

\(\frac{x-1}{2}=5\Rightarrow x=11;\)\(\frac{y-2}{3}=5\Rightarrow y=17;\)\(\frac{z-3}{4}=5\Rightarrow z=23.\)