document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198615>Question 1198615</A>: <I><SPAN STYLE=\"word-break: break-all;\">   The ABC Ink Company is a small family owned company that sells packages of ink cartridge refills <BR>\n" );
document.write( "for smartpens.  The Xavier set contains one blue ink refill and one black ink refill.  The Yvonne set <BR>\n" );
document.write( "includes two blue ink refills, three black ink refills, and one red ink refill.  The Zena set includes <BR>\n" );
document.write( "four blue ink refills, five black ink refills, and one red ink refill.  The company has sold most of its <BR>\n" );
document.write( "stock and has found that it has only 11 blue ink cartridge refills, 14 black ink cartridge refills, and 3 <BR>\n" );
document.write( "red ink cartridge refills.  How many of each set should the company package to sell in order to use <BR>\n" );
document.write( "all of the remaining ink cartridges so that there will be none left in inventory. <BR>\n" );
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document.write( "a.  Describe what the variables represent:  =\r<BR>\n" );
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document.write( "b.  Write the system of linear equations: <BR>\n" );
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document.write( "c.  Solve the system of equations using Gauss-Jordan Elimination.  Show all proper row operations <BR>\n" );
document.write( "and the resulting matrices.  Write the solution as an ordered triple, if appropriate. <BR>\n" );
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document.write( "d.  Fill out the table to show the possible combinations of sets the company can package: <BR>\n" );
document.write( "       x               y              z <BR>\n" );
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document.write( "e.  In at least one complete sentence with proper grammar and correct spelling, write the  <BR>\n" );
document.write( "solutions in terms of what the variables represent.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848250 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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=848250\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a. Define Variables:**\r<BR>\n" );
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document.write( "* **x:** Number of Xavier sets to be packaged.<BR>\n" );
document.write( "* **y:** Number of Yvonne sets to be packaged.<BR>\n" );
document.write( "* **z:** Number of Zena sets to be packaged.\r<BR>\n" );
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document.write( "**b. System of Linear Equations:**\r<BR>\n" );
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document.write( "* **Blue Ink Refills:** 1x + 2y + 4z = 11 <BR>\n" );
document.write( "* **Black Ink Refills:** 1x + 3y + 5z = 14<BR>\n" );
document.write( "* **Red Ink Refills:** 0x + 1y + 1z = 3 \r<BR>\n" );
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document.write( "**c. Gauss-Jordan Elimination**\r<BR>\n" );
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document.write( "1. **Augmented Matrix:**\r<BR>\n" );
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document.write( "   ```<BR>\n" );
document.write( "   [ 1  2  4 | 11 ]<BR>\n" );
document.write( "   [ 1  3  5 | 14 ]<BR>\n" );
document.write( "   [ 0  1  1 |  3 ]<BR>\n" );
document.write( "   ```\r<BR>\n" );
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document.write( "2. **Row Operations:**\r<BR>\n" );
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document.write( "   * **R2 = R2 - R1:** <BR>\n" );
document.write( "      ```<BR>\n" );
document.write( "      [ 1  2  4 | 11 ]<BR>\n" );
document.write( "      [ 0  1  1 |  3 ]<BR>\n" );
document.write( "      [ 0  1  1 |  3 ]<BR>\n" );
document.write( "      ```\r<BR>\n" );
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document.write( "   * **R3 = R3 - R2:** <BR>\n" );
document.write( "      ```<BR>\n" );
document.write( "      [ 1  2  4 | 11 ]<BR>\n" );
document.write( "      [ 0  1  1 |  3 ]<BR>\n" );
document.write( "      [ 0  0  0 |  0 ]<BR>\n" );
document.write( "      ```\r<BR>\n" );
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document.write( "   * **R1 = R1 - 2R2:** <BR>\n" );
document.write( "      ```<BR>\n" );
document.write( "      [ 1  0  2 |  5 ]<BR>\n" );
document.write( "      [ 0  1  1 |  3 ]<BR>\n" );
document.write( "      [ 0  0  0 |  0 ]<BR>\n" );
document.write( "      ```\r<BR>\n" );
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document.write( "3. **Reduced Row-Echelon Form:**\r<BR>\n" );
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document.write( "   ```<BR>\n" );
document.write( "      [ 1  0  2 |  5 ]<BR>\n" );
document.write( "      [ 0  1  1 |  3 ]<BR>\n" );
document.write( "      [ 0  0  0 |  0 ]<BR>\n" );
document.write( "      ```\r<BR>\n" );
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document.write( "**d. Possible Combinations of Sets**\r<BR>\n" );
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document.write( "* From the reduced row-echelon form:<BR>\n" );
document.write( "    * x + 2z = 5 <BR>\n" );
document.write( "    * y + z = 3\r<BR>\n" );
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document.write( "* **Possible Solutions:**<BR>\n" );
document.write( "    * Let z = 0: <BR>\n" );
document.write( "        * x = 5, y = 3<BR>\n" );
document.write( "    * Let z = 1: <BR>\n" );
document.write( "        * x = 3, y = 2<BR>\n" );
document.write( "    * Let z = 2: <BR>\n" );
document.write( "        * x = 1, y = 1 \r<BR>\n" );
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document.write( "* **Table of Possible Combinations:**\r<BR>\n" );
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document.write( "   | x (Xavier) | y (Yvonne) | z (Zena) |<BR>\n" );
document.write( "   |---|---|---|<BR>\n" );
document.write( "   | 5 | 3 | 0 |<BR>\n" );
document.write( "   | 3 | 2 | 1 |<BR>\n" );
document.write( "   | 1 | 1 | 2 |\r<BR>\n" );
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document.write( "**e. Solution Interpretation:**\r<BR>\n" );
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document.write( "The company can package the remaining ink cartridges in three possible ways: \r<BR>\n" );
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document.write( "* 5 Xavier sets, 3 Yvonne sets, and 0 Zena sets.<BR>\n" );
document.write( "* 3 Xavier sets, 2 Yvonne sets, and 1 Zena set.<BR>\n" );
document.write( "* 1 Xavier set, 1 Yvonne set, and 2 Zena sets. \r<BR>\n" );
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document.write( "This will ensure all 11 blue, 14 black, and 3 red ink cartridges are used.<BR>\n" );
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