Question 1017226
<pre>
Let the number of dimes be x
Let the number of quarters be y


                      Value      Value
Type       Number       of         of
 of          of        EACH       ALL
coin        coins      coin      coins
-------------------------------------------
dimes         x      $0.10     $0.10x
quarters      y      $0.25     $0.25y
-------------------------------------------
TOTALS       30      -----     $4.65

 The first equation comes from the second column.

  {{{(matrix(3,1,Number,of,dimes))}}}{{{""+""}}}{{{(matrix(3,1,Number,of,quarters))}}}{{{""=""}}}{{{(matrix(4,1,total,number,of,coins))}}}

                   x + y = 30

 The second equation comes from the last column.
  {{{(matrix(4,1,Value,of,ALL,dimes))}}}{{{""+""}}}{{{(matrix(4,1,Value,of,ALL,quarters))}}}{{{""=""}}}{{{(matrix(5,1,Total,value,of,ALL,coins))}}}

           0.1x + 0.25y = 4.65

Get rid of decimals by multiplying every term by 100:

              10x + 25y = 465

 So we have the system of equations:
           {{{system(x + y = 30,10x + 25y = 465)}}}.

We solve by substitution.  Solve the first equation for y:

              x + y = 30
                  y = 30 - x

Substitute (30 - x) for y in 10x + 25y = 465

   10x + 25(30 - x) = 465
    10x + 750 - 25x = 465
         -15x + 750 = 465
               -15x = -285
                  x = 19 = the number of dimes.

Substitute in y = 30 - x
              y = 30 - (19
              y = 11 quarters.

Checking:  19 dimes is $1.90 and 11 quarters is $2.75
            That's 30 coins.
            And indeed $1.90 + $2.75 = $4.65
Edwin</pre>