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A chemical company makes two brands of antifreeze. The first brand is 60% pure antifreeze, and the second brand is 85% pure antifreeze. In order to obtain 110 gallons of a mixture that contains 75% pure antifreeze, how many gallons of each brand of antifreeze must be used

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Let x = amount of 45% antifreeze

Let y = amount of 70% antifreeze

   EQUATION 1:   x + y = 150    (total of 150 gallons mixed)

   EQUATION 2:  .45x + .75y = .55(x + y)

Simplify and solve the system of equations

   Multiply second equation by 100 on both sides to remove the decimals

         45x + 75y = 55(x + y)

   Combine like terms

         45x + 75y = 55x + 55y

         45x - 55x + 75y - 55y = 0

 

         -10x + 20y = 0

     Now we have the following system of equations:

            x  +    y = 150

       -10x + 20y =     0

   Multiply the first equation by -10 to get opposite coefficients for x;  add the equations to eliminate x

         10x + 10y = 1500

        -10x + 20y =       0

       ------------------------------

                   30y = 1500

    Solve for y

          30y = 1500

              y = 50

    Since the total mixed gallons is 150, x = 150 - 50 = 100

So we need 100 gallons of the 45% antifreeze and 50 gallons of the 70% antifreeze

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