Question 18: Find x, y, z knowing \(\frac{x}{{11}} = \frac{y}{{ – 3}} = \frac{z}{{ – 4}};x + y + z = 12.\)

\(\begin{aligned}&\frac{x}{{11}} = \frac{y}{{ – 3}} = \frac{z}{{ – 4}};x + y + z = 12 \\&\text{Applying the property of the sequence of equal ratios we have:}\\ &\frac{x}{{11}} = \frac{y}{{ – 3}} = \frac{z} {{ – 4}} = \frac{{x + y + z}}{{11 + \left( { – 3} \right) + \left( { – 4} \right)}} = \frac{{ 12}}{4} = 3\\&\frac{x}{{11}} = 3 \Rightarrow x = 3.11 \Rightarrow x = 33\\&\frac{y}{{ – 3}} = 3 \ Rightarrow y = 3.\left( { – 3} \right) \Rightarrow y = – 9\\&\frac{z}{-4} = 3 \Rightarrow z = 3.(-4) \Rightarrow z = – 12\\&\text{So }x=33, y=-9, z=-12.\end{aligned} \)