Journey

The road from A to B measures 11.5 km. Firstly, up the hill, then level plane, and then downhill. Tourists go uphill at 3 km/h, on planes at 4 km/h, and downhill at 5 km/h. From point A to B, it went 2 hours, 54 minutes back, 3 hours, and 6 minutes. How long is the segment of the level plane?

Correct answer:

y =  4 km

Step-by-step explanation:

$$\begin{gathered} s=11.5\text{km} \\ {t}_1=2:54=2hr54min=2+\frac{54}{60}=2.9hr=2.9 \\ {t}_2=3:6=3hr6min=3+\frac{6}{60}=3.1hr=3.1 \\ {v}_1=3\text{km/h} \\ {v}_2=4\text{km/h} \\ {v}_3=5\text{km/h} \\ x+y+z=s \\ x/v_1+y/v_2+z/v_3=t_1 \\ x/v_3+y/v_2+z/v_1=t_2 \\ x+y+z=11.5 \\ x/3+y/4+z/5=2.9 \\ x/5+y/4+z/3=3.1 \\ x+y+z=11.5 \\ 20x+15y+12z=174 \\ 12x+15y+20z=186 \\ Pivot:Row1\leftrightarrow Row2 \\ 20x+15y+12z=174 \\ x+y+z=11.5 \\ 12x+15y+20z=186 \\ Row2-\frac{1}{20}\cdot Row1\to Row2 \\ 20x+15y+12z=174 \\ 0.25y+0.4z=2.8 \\ 12x+15y+20z=186 \\ Row3-\frac{12}{20}\cdot Row1\to Row3 \\ 20x+15y+12z=174 \\ 0.25y+0.4z=2.8 \\ 6y+12.8z=81.6 \\ Pivot:Row2\leftrightarrow Row3 \\ 20x+15y+12z=174 \\ 6y+12.8z=81.6 \\ 0.25y+0.4z=2.8 \\ Row3-\frac{0.25}{6}\cdot Row2\to Row3 \\ 20x+15y+12z=174 \\ 6y+12.8z=81.6 \\ -0.133z=-0.6 \\ z=\frac{-0.6}{-0.13333333}=4.5 \\ y=\frac{81.6-12.8z}{6}=\frac{81.6-12.8\cdot 4.5}{6}=4=4\text{km} \\ x=\frac{174-15y-12z}{20}=\frac{174-15\cdot 4-12\cdot 4.5}{20}=3 \\ x=3 \\ y=4 \\ z=\frac{9}{2}=4.5 \end{gathered}$$

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