MHB Using linear systems to solve problems (3)

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Rajah's earnings consist of an hourly wage and tips. In the first week, he worked 12 hours and earned a total of $117. In the second week, he worked 10 hours and earned $110, with tips equal to the previous week. By setting up a system of linear equations based on these earnings, one can solve for Rajah's hourly wage. The solution reveals the hourly wage Rajah earns from his work.
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Rajah earns an hourly wage plus tips. One week he worked 12 h and made a total of \$117. The next week he worked 10 h and earned the same amount in tips as the week before, for a total of \$110. What is Rajah's hourly wage?
 
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Only the slightest variation from the rental car problem.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

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