Question

a plane flying with the wind travels 630 mi in 1.5hrs. Flying against the wind , the plane travels 1120 mi in 4hrs. Find the rate of the plane in still air and wind speed.

Answer 1

$\bf \begin{cases} r=\textit{rate of the plane}\\ w=\textit{rate of the wind}\\ \end{cases} \\\\ \begin{array}{ccccllll} &distance&rate&time(hrs)\\ &\textendash\textendash\textendash\textendash\textendash\textendash&\textendash\textendash\textendash\textendash\textendash\textendash&\textendash\textendash\textendash\textendash\textendash\textendash\textendash\textendash\textendash\\ \textit{with wind}&630&r+w&1\frac{1}{2}\\ \textit{against wind}&1120&r-w&4 \end{array}$

$\bf thus \begin{cases} 630=(r+w)1\frac{1}{2}\to 630=(r+w)\frac{3}{2} \\\\ 1120=(r-w)4\\ --------------\\ 630=(r+w)\frac{3}{2}\implies 630\cdot \frac{2}{3}=r+w\\ 420=r+w\implies \boxed{420-w}=r\\ --------------\\ thus\\ --------------\\ 1120=(r-w)4\implies 1120=4r-4w \\\\ 1120=4(\boxed{420-w})-4w \end{cases}$

solve for "w", to find the wind's speed rate,

so hmmm what's the plane's rate?  well 420 - w = r  :)