Topic 5.2      (Problem Solving)

The following problems will be solved with a system of two equations.

Sample Problem

The different between two numbers is 9.  The sum of twice the larger number and

   three times the smaller number is 33.  Find the two numbers.

       x    =    smaller number           Read the problem first, and then use variables

       y    =    larger number              to represent the unknown.  (called the legend)

      y    -    x    =    9               The first equation

    2y   +   3x   =   33             The second equation - Multiply the first equation by 3

    3y   -   3x   =   27              and add to the second equation.

    2y   +  3x   =   33

    5y              =   60

       y     =    12      The larger number

    Substitute 12 into the first equation:   12   -   x   =   9,   -x   =   -3,   x   =   3

       x   =   3           The smaller number

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Donald collected $16.35 in dimes and quarters from a vending machine.  If he had

  a total of 87 coins, how many quarters and dimes did he get?

    Let x   =   number of quarters

    Let y   =   number of dimes

          x    +       y    =         87          First equation

      25x    +  10y    =    1635           Second equation - It is easier to use no decimal and

                                                          put all numbers in terms cents.

      Multiply the first equation by -10.

     -10x    -    10y    =    -870

      25x    +   10y    =    1635

              15x           =       765

                         x    =    51                         Number of quarters

        51    +    y    =    87,      y    =   36    Number of dimes      Answers

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Cara is making punch for a party.  She wants to combine a 20% real fruit juice

   drink with a 100% real fruit juice drink to get 16 cups of a mixture that is 80%

   real fruit juice.  How much of each type should she use?

                                Make a chart like the following.

                                 No. of cups                  %                     Amount of pure fruit juice

                    20%              x                            .2                                             .2x

                   100%             y                             1                                                y

Mixture       80%            16                           .8                                            12.8

The (%) column is the per cent changed to a decimal.

The last column is the first column multiplied by the second column.

The first equation comes from the first column and the second equation

  comes from the third column

      x    +    y    =    16      Multiply the second equation by 10 to get rid of the

   .2x    +    y   =   12.8     decimal.

     -2x    -    2y    =    -32    Multiply the first equation by -2 and add to the

      2x    +  10y   =    128   second equation.

                     8y   =      96

                       y   =      12            Substitute y into the first equation.                       

                                                     x    +    12    =    16,    x    =    4   

                 4   cups   @   20%

               12   cups   @   100%      Answers