Question 208212

SOLUTION:

Let x = number of nickels
    y = number of dimes
    z = number of quarters



Write equations from the information given in the word problem:

a. A coin collector has 41 coins consisting of nickels, dimes and quarters...:

               x + y + z = 41   -->Equation 1

b. ...and they are worth a total of $4.00:

    0.05x + 0.1y + 0.25z = 4.00 -->Equation 2

c. ...the number of dimes plus the number of quarters is one more than the number of nickels...:

                   y + z = x + 1
               x - y - z = -1    -->Equation 3


Now we have a system of 3 linear equations with 3 unknowns:

            x +    y +     z =  41  -->Equation 1
        0.05x + 0.1y + 0.25z =  4   -->Equation 2
            x -    y -     z = -1   -->Equation 3



The augmented 3x3 matrix for the above system of equations is:

                |   1   1    1 | 41 |
                | .05  .1  .25 |  4 |
                |   1  -1   -1 | -1 |


Using Cramer's Rule:

{{{x = Dx/D}}}
{{{y = Dy/D}}}
{{{z = Dz/D}}}



First we solve for D, Dx ,Dy and Dz:


D =  |   1   1    1 |   
     | .05  .1  .25 | 
     |   1  -1   -1 | 

D = 1(-.1+.25) - .05(-1+1) + 1(.25-.1)   

D = 1(.15) - .05(0) + 1(.15)

D = .15 - 0 - .15

D = 0 

We cannot solve using Determinants because D = 0 so try other methods.


Multiply Equation 2 by 4 and add to Equation 3 to eliminate z:


         [  0.05x + 0.1y + 0.25z =  4 ] * 4 -->4*Equation 2


                 .2x  +.4y + z = 16   -->Equation 4
                   x  -  y - z = -1   -->Equation 3
                ___________________
                    1.2x - .6y = 15   -->Equation 5


Equation 1 + Equation 3:


            x +    y +     z =  41  -->Equation 1
            x -    y -     z =  -1  -->Equation 3
           _______________________
            2x               =  40  -->Equation 6


                             2x = 40
                             {{{x = 40/2}}}
                             {{{x = 20}}}


Substitute x = 20 in Equation 5:

                     1.2(20) - .6y = 15 
                     24 - .6y = 15
                     24 - 15  = .6y
                           9  = .6y
                           y  = 9/.6
                           {{{y  = 15}}}


Substitute {{{x}}}  and {{{y}}} in Equation 1:


                  20 + 15 + z =  41  
                       35 + z = 41
                            z = 41-35
                            {{{z = 6}}}


ANSWER: The collector has 20 nickels, 15 dimes and 6 quarters.