Question 1160712
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Let x be her speed driving to Santa Barbara, in miles per hour.

Then her speed driving back is (x-10) mph, according to the condition.



The time driving "to there" is  {{{200/x}}} hours  (the distance divided by speed).

The time driving back is  {{{200/(x-10)}}} hours.



Time back is one hour longer than the time "to there".

It gives you THIS EQUATION

    {{{200/(x-10)}}} - {{{200/x}}} = 1  hour.



It is the major step of the solution to establish this equation.

It is called "time" equation, since each its term is the time.



From this point, I just see / guess the solution mentally:  x = 50 miles per hour.



To solve the equation formally, multiply both sides by x*(x-10).  You will get


    200x - 200*(x-10) = x*(x-10).


Simplify and solve for x

    2000 = x^2 - 10x,

    x^2 - 10x - 1000 = 0.

Factor left side

    (x-50)*(x+40) = 0.



The solution is  x= 50 mph  (exactly as I guessed above).


<U>CHECK</U>.  Time to Santa Barbara is  {{{200/50}}} = 4 hours.

        Time driving back is  {{{200(50-10)}}} = {{{200/40}}} = 5 hourrs.

        The difference is  5 hours - 4 hours = 1 hour.    ! Precisely correct !
</pre>

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