Journey to spa

Mr. Dvorak took his wife to the spa, and the journey took 3.5 hours. On the way back, he drove 12 km/h faster and was home in 3 hours. At what speed did he go to the spa, and at what speed did he return?

Correct answer:

v₁ =  72 km/h
v₂ =  84 km/h

Step-by-step explanation:

$$\begin{gathered} s=v_1\cdot t_1=v_2\cdot t_2 \\ {t}_1=3.5\text{h} \\ {t}_2=3\text{h} \\ {v}_1\cdot t_1=v_2\cdot t_2 \\ {v}_2=v_1+12 \\ {v}_1\cdot 3.5=v_2\cdot 3 \\ {v}_2=v_1+12 \\ 3.5v_1-3v_2=0 \\ {v}_1-v_2=-12 \\ Row2-\frac{1}{3.5}\cdot Row1\to Row2 \\ 3.5v_1-3v_2=0 \\ -0.14v_2=-12 \\ {v}_2=\frac{-12}{-0.14285714}=84 \\ {v}_1=\frac{0+3v_2}{3.5}=\frac{0+3\cdot 84}{3.5}=72=72\text{km/h} \\ {v}_1=72 \\ {v}_2=84 \end{gathered}$$

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