Question 12: Find x, y, z knowing \(\frac{x}{{ – 2}} = \frac{y}{3} = \frac{z}{5};x + y + z = – 12 .\)

\(\begin{aligned}&\frac{x}{{ – 2}} = \frac{y}{3} = \frac{z}{5};x + y + z = – 12\\&\ text{Applying the property of the sequence of equal ratios we have:}\\ &\frac{x}{{ – 2}} = \frac{y}{3} = \frac{z}{5} = \frac {{x + y + z}}{{ – 2 + 3 + 5}} = \frac{{ – 12}}{6} = – 2\\&\frac{x}{{ – 2}} = – 2 \Rightarrow x = \left( { – 2} \right).\left( { – 2} \right) \Rightarrow x = 4\\&\frac{y}{3} = – 2 \Rightarrow y = \ left( { – 2} \right).3 \Rightarrow y = – 6\\&\frac{z}{5} = – 2 \Rightarrow z = \left( { – 2} \right).5 \Rightarrow z = – 10\\&\text{So }x=4, y=-6, z=-10.\end{aligned} \)