document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=208212>Question 208212</A>: <I><SPAN STYLE=\"word-break: break-all;\"> SOLVE USING DETERMINANTS and CRAMER'S RULE\r<BR>\n" );
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document.write( "A coin collector has 41 coins consisting of nickels, dimes and quarters and they are worth a total of $4.00. If the number of dimes plus the number of quarters is one more than the number of nickels, then how many of each does he have ?\r<BR>\n" );
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document.write( "REQUIRED: Find the number of quarters, dimes and nickels. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #157513 by </B><A HREF=/tutors/aboutme.mpl?userid=pjalgeb><B>pjalgeb(9)</B></A><img src=\"/images/stars/stars-05.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=pjalgeb><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=157513\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "SOLUTION:\r<BR>\n" );
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document.write( "Let x = number of nickels<BR>\n" );
document.write( "    y = number of dimes<BR>\n" );
document.write( "    z = number of quarters\r<BR>\n" );
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document.write( "Write equations from the information given in the word problem:\r<BR>\n" );
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document.write( "a. A coin collector has 41 coins consisting of nickels, dimes and quarters...:\r<BR>\n" );
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document.write( "               x + y + z = 41   -->Equation 1\r<BR>\n" );
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document.write( "b. ...and they are worth a total of $4.00:\r<BR>\n" );
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document.write( "    0.05x + 0.1y + 0.25z = 4.00 -->Equation 2\r<BR>\n" );
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document.write( "c. ...the number of dimes plus the number of quarters is one more than the number of nickels...:\r<BR>\n" );
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document.write( "                   y + z = x + 1<BR>\n" );
document.write( "               x - y - z = -1    -->Equation 3\r<BR>\n" );
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document.write( "Now we have a system of 3 linear equations with 3 unknowns:\r<BR>\n" );
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document.write( "            x +    y +     z =  41  -->Equation 1<BR>\n" );
document.write( "        0.05x + 0.1y + 0.25z =  4   -->Equation 2<BR>\n" );
document.write( "            x -    y -     z = -1   -->Equation 3\r<BR>\n" );
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document.write( "The augmented 3x3 matrix for the above system of equations is:\r<BR>\n" );
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document.write( "                |   1   1    1 | 41 |<BR>\n" );
document.write( "                | .05  .1  .25 |  4 |<BR>\n" );
document.write( "                |   1  -1   -1 | -1 |\r<BR>\n" );
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document.write( "Using Cramer's Rule:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+Dx%2FD ALIGN=MIDDLE  ALT=\"x+=+Dx%2FD\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+Dy%2FD ALIGN=MIDDLE  ALT=\"y+=+Dy%2FD\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+Dz%2FD ALIGN=MIDDLE  ALT=\"z+=+Dz%2FD\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "First we solve for D, Dx ,Dy and Dz:\r<BR>\n" );
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document.write( "D =  |   1   1    1 |   <BR>\n" );
document.write( "     | .05  .1  .25 | <BR>\n" );
document.write( "     |   1  -1   -1 | \r<BR>\n" );
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document.write( "D = 1(-.1+.25) - .05(-1+1) + 1(.25-.1)   \r<BR>\n" );
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document.write( "D = 1(.15) - .05(0) + 1(.15)\r<BR>\n" );
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document.write( "D = .15 - 0 - .15\r<BR>\n" );
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document.write( "D = 0 \r<BR>\n" );
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document.write( "We cannot solve using Determinants because D = 0 so try other methods.\r<BR>\n" );
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document.write( "Multiply Equation 2 by 4 and add to Equation 3 to eliminate z:\r<BR>\n" );
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document.write( "         [  0.05x + 0.1y + 0.25z =  4 ] * 4 -->4*Equation 2\r<BR>\n" );
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document.write( "                 .2x  +.4y + z = 16   -->Equation 4<BR>\n" );
document.write( "                   x  -  y - z = -1   -->Equation 3<BR>\n" );
document.write( "                ___________________<BR>\n" );
document.write( "                    1.2x - .6y = 15   -->Equation 5\r<BR>\n" );
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document.write( "Equation 1 + Equation 3:\r<BR>\n" );
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document.write( "            x +    y +     z =  41  -->Equation 1<BR>\n" );
document.write( "            x -    y -     z =  -1  -->Equation 3<BR>\n" );
document.write( "           _______________________<BR>\n" );
document.write( "            2x               =  40  -->Equation 6\r<BR>\n" );
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document.write( "                             2x = 40<BR>\n" );
document.write( "                             <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+40%2F2 ALIGN=MIDDLE  ALT=\"x+=+40%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "                             <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+20 ALIGN=MIDDLE  ALT=\"x+=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Substitute x = 20 in Equation 5:\r<BR>\n" );
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document.write( "                     1.2(20) - .6y = 15 <BR>\n" );
document.write( "                     24 - .6y = 15<BR>\n" );
document.write( "                     24 - 15  = .6y<BR>\n" );
document.write( "                           9  = .6y<BR>\n" );
document.write( "                           y  = 9/.6<BR>\n" );
document.write( "                           <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y++=+15 ALIGN=MIDDLE  ALT=\"y++=+15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > in Equation 1:\r<BR>\n" );
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document.write( "                  20 + 15 + z =  41  <BR>\n" );
document.write( "                       35 + z = 41<BR>\n" );
document.write( "                            z = 41-35<BR>\n" );
document.write( "                            <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+6 ALIGN=MIDDLE  ALT=\"z+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "ANSWER: The collector has 20 nickels, 15 dimes and 6 quarters.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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