Question 1132844
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The system of equations is

    10x + 25y = 690   cents     (1)   (x = # of dimes; y = # of quarters)

    25x + 10y = 780   cents     (2)


To solve the system, multiply equation (1) by 5 (both sides) and multiply equation (2) by 2. You will get


    50x + 125y = 5*690           (3)

    50x + 20y = 2*780            (4)


Now subtract eq(4) from eq(3). The terms "50x" will cancel each other, so you eliminate the unknown "x" and get
a single equation for the unknown y in this way (it is how the Elimination method works)

          125y - 20y = 5*690 - 2*780

          105y = 1890     ==============>  y = {{{1890/105}}} = 18.


<U>ANSWER</U>.  The number of quarters in the bank is 18.
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Solved.


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On the way, you learned from my post on how the Elimination method works.