document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178756>Question 1178756</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A natural history museum borrows $2,000,000 at simple annual interest to purchase new exhibits. Some of the money is borrowed at 5%, some at 6.5%, and some at 7.5%. Use a system of linear equations to determine how much is borrowed at each rate given that the total annual interest is $121,250 and the amount borrowed at 6.5% is four times the amount borrowed at 7.5%. Solve the system of linear equations using matrices.\r<BR>\n" );
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document.write( "at 5%<BR>\n" );
document.write( "at 6.5%<BR>\n" );
document.write( "at 6.5% </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850300 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850300\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's set up the system of linear equations and solve it using matrices.\r<BR>\n" );
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document.write( "**1. Define the Variables:**\r<BR>\n" );
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document.write( "* x = amount borrowed at 5%<BR>\n" );
document.write( "* y = amount borrowed at 6.5%<BR>\n" );
document.write( "* z = amount borrowed at 7.5%\r<BR>\n" );
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document.write( "**2. Formulate the Equations:**\r<BR>\n" );
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document.write( "* **Total Amount Borrowed:** x + y + z = 2,000,000<BR>\n" );
document.write( "* **Total Annual Interest:** 0.05x + 0.065y + 0.075z = 121,250<BR>\n" );
document.write( "* **Relationship Between y and z:** y = 4z\r<BR>\n" );
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document.write( "**3. Rewrite the Equations in Standard Form:**\r<BR>\n" );
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document.write( "1.  x + y + z = 2,000,000<BR>\n" );
document.write( "2.  0.05x + 0.065y + 0.075z = 121,250<BR>\n" );
document.write( "3.  0x + y - 4z = 0\r<BR>\n" );
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document.write( "To simplify the second equation, multiply it by 1000:\r<BR>\n" );
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document.write( "1.  x + y + z = 2,000,000<BR>\n" );
document.write( "2.  50x + 65y + 75z = 121,250,000<BR>\n" );
document.write( "3.  y - 4z = 0\r<BR>\n" );
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document.write( "**4. Represent the System as an Augmented Matrix:**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "[ 1   1   1 | 2000000 ]<BR>\n" );
document.write( "[ 50  65  75 | 121250000 ]<BR>\n" );
document.write( "[ 0   1  -4 | 0 ]<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**5. Solve Using Row Operations:**\r<BR>\n" );
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document.write( "1.  **Eliminate x from the second row:**<BR>\n" );
document.write( "    * R2 = R2 - 50 * R1<BR>\n" );
document.write( "    ```<BR>\n" );
document.write( "    [ 1   1   1 | 2000000 ]<BR>\n" );
document.write( "    [ 0  15  25 | 21250000 ]<BR>\n" );
document.write( "    [ 0   1  -4 | 0 ]<BR>\n" );
document.write( "    ```\r<BR>\n" );
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document.write( "2.  **Swap R2 and R3 to make R2 easier to work with:**<BR>\n" );
document.write( "    ```<BR>\n" );
document.write( "    [ 1   1   1 | 2000000 ]<BR>\n" );
document.write( "    [ 0   1  -4 | 0 ]<BR>\n" );
document.write( "    [ 0  15  25 | 21250000 ]<BR>\n" );
document.write( "    ```\r<BR>\n" );
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document.write( "3.  **Eliminate y from the third row:**<BR>\n" );
document.write( "    * R3 = R3 - 15 * R2<BR>\n" );
document.write( "    ```<BR>\n" );
document.write( "    [ 1   1   1 | 2000000 ]<BR>\n" );
document.write( "    [ 0   1  -4 | 0 ]<BR>\n" );
document.write( "    [ 0   0  85 | 21250000 ]<BR>\n" );
document.write( "    ```\r<BR>\n" );
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document.write( "4.  **Solve for z:**<BR>\n" );
document.write( "    * 85z = 21,250,000<BR>\n" );
document.write( "    * z = 21,250,000 / 85<BR>\n" );
document.write( "    * z = 250,000\r<BR>\n" );
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document.write( "5.  **Solve for y:**<BR>\n" );
document.write( "    * y - 4z = 0<BR>\n" );
document.write( "    * y = 4z<BR>\n" );
document.write( "    * y = 4 * 250,000<BR>\n" );
document.write( "    * y = 1,000,000\r<BR>\n" );
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document.write( "6.  **Solve for x:**<BR>\n" );
document.write( "    * x + y + z = 2,000,000<BR>\n" );
document.write( "    * x = 2,000,000 - y - z<BR>\n" );
document.write( "    * x = 2,000,000 - 1,000,000 - 250,000<BR>\n" );
document.write( "    * x = 750,000\r<BR>\n" );
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document.write( "**6. Answers:**\r<BR>\n" );
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document.write( "* Amount borrowed at 5% (x): $750,000<BR>\n" );
document.write( "* Amount borrowed at 6.5% (y): $1,000,000<BR>\n" );
document.write( "* Amount borrowed at 7.5% (z): $250,000<BR>\n" );
document.write( " </SPAN>\n" );
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