document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=111380>Question 111380</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Part of $20,000 is invested at 5% and the rest at 3%. How much is invested at each rate if the total income is 4.5% of the total investment? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #81223 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=81223\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x=amount invested for 5%, y=amount for 3%\r<BR>\n" );
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document.write( "So we know the sum of the parts is the total $20,000. So this means \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=20000 ALIGN=MIDDLE  ALT=\"x%2By=20000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Also we know that 5% of x amount of dollars plus 3% of y amount of dollars is 4.5% of $20,000 (which is $900)\r<BR>\n" );
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document.write( "So \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.05x%2B0.03y=900 ALIGN=MIDDLE  ALT=\"0.05x%2B0.03y=900\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%2B3y=90000 ALIGN=MIDDLE  ALT=\"5x%2B3y=90000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by 100 to make every number a whole number\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=solving_linear_system_by_elimination.solver>Solving a System of Linear Equations by Elimination/Addition</A></TD></TR>\n" );
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document.write( "  Lets start with the given system of linear equations<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B1%2Ay=20000 ALIGN=MIDDLE  ALT=\"1%2Ax%2B1%2Ay=20000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax%2B3%2Ay=90000 ALIGN=MIDDLE  ALT=\"5%2Ax%2B3%2Ay=90000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).<BR>\n" );
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document.write( "  So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.<BR>\n" );
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document.write( "  So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and 5 to some equal number, we could try to get them to the LCM.<BR>\n" );
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document.write( "  Since the LCM of 1 and 5 is 5, we need to multiply both sides of the top equation by 5 and multiply both sides of the bottom equation by -1 like this:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2A%281%2Ax%2B1%2Ay%29=%2820000%29%2A5 ALIGN=MIDDLE  ALT=\"5%2A%281%2Ax%2B1%2Ay%29=%2820000%29%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the top equation (both sides) by 5<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2A%285%2Ax%2B3%2Ay%29=%2890000%29%2A-1 ALIGN=MIDDLE  ALT=\"-1%2A%285%2Ax%2B3%2Ay%29=%2890000%29%2A-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the bottom equation (both sides) by -1<BR>\n" );
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document.write( "  So after multiplying we get this:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax%2B5%2Ay=100000 ALIGN=MIDDLE  ALT=\"5%2Ax%2B5%2Ay=100000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2Ax-3%2Ay=-90000 ALIGN=MIDDLE  ALT=\"-5%2Ax-3%2Ay=-90000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Notice how 5 and -5 add to zero (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2B-5=0 ALIGN=MIDDLE  ALT=\"5%2B-5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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document.write( "  Now add the equations together. In order to add 2 equations, group like terms and combine them<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2Ax-5%2Ax%29%2B%285%2Ay-3%2Ay%29=100000-90000 ALIGN=MIDDLE  ALT=\"%285%2Ax-5%2Ax%29%2B%285%2Ay-3%2Ay%29=100000-90000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285-5%29%2Ax%2B%285-3%29y=100000-90000 ALIGN=MIDDLE  ALT=\"%285-5%29%2Ax%2B%285-3%29y=100000-90000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%285%2B-5%29%2Ax%2B%285-3%29%2Ay=100000-90000 ALIGN=MIDDLE  ALT=\"cross%285%2B-5%29%2Ax%2B%285-3%29%2Ay=100000-90000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.<BR>\n" );
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document.write( "  So after adding and canceling out the x terms we're left with:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ay=10000 ALIGN=MIDDLE  ALT=\"2%2Ay=10000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=10000%2F2 ALIGN=MIDDLE  ALT=\"y=10000%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to solve for y<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5000 ALIGN=MIDDLE  ALT=\"y=5000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
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document.write( "  Now plug this answer into the top equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B1%2Ay=20000 ALIGN=MIDDLE  ALT=\"1%2Ax%2B1%2Ay=20000\"   DISABLED_style= \"max-width:90%; height:auto;\" > to solve for x<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B1%285000%29=20000 ALIGN=MIDDLE  ALT=\"1%2Ax%2B1%285000%29=20000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5000 ALIGN=MIDDLE  ALT=\"y=5000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B5000=20000 ALIGN=MIDDLE  ALT=\"1%2Ax%2B5000=20000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax=20000-5000 ALIGN=MIDDLE  ALT=\"1%2Ax=20000-5000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5000 ALIGN=MIDDLE  ALT=\"5000\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax=15000 ALIGN=MIDDLE  ALT=\"1%2Ax=15000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the terms on the right side<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%28%281%2F1%29%281%29%29%2Ax=%2815000%29%281%2F1%29 ALIGN=MIDDLE  ALT=\"cross%28%281%2F1%29%281%29%29%2Ax=%2815000%29%281%2F1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F1 ALIGN=MIDDLE  ALT=\"1%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This will cancel out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the left side.<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=15000 ALIGN=MIDDLE  ALT=\"x=15000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the terms on the right side<BR>\n" );
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document.write( "  So our answer is <BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=15000 ALIGN=MIDDLE  ALT=\"x=15000\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5000 ALIGN=MIDDLE  ALT=\"y=5000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  which also looks like<BR>\n" );
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document.write( "  (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15000 ALIGN=MIDDLE  ALT=\"15000\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5000 ALIGN=MIDDLE  ALT=\"5000\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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document.write( "  Notice if we graph the equations (if you need help with graphing, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver</a>)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B1%2Ay=20000 ALIGN=MIDDLE  ALT=\"1%2Ax%2B1%2Ay=20000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax%2B3%2Ay=90000 ALIGN=MIDDLE  ALT=\"5%2Ax%2B3%2Ay=90000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  we get <BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A++drawing%28+500%2C+600%2C+-15010%2C+15010%2C+-5010%2C+5010%2C%0D%0A++++graph%28+500%2C+600%2C+-15010%2C+15010%2C+-5010%2C+5010%2C+%2820000-1%2Ax%29%2F1%2C+%2890000-5%2Ax%29%2F3+%29%2C%0D%0A++++blue%28++circle%28+15000%2C+5000%2C+200.133333333333+%29+%29+%0D%0A++%29%0D%0A++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B1%2Ay=20000 ALIGN=MIDDLE  ALT=\"1%2Ax%2B1%2Ay=20000\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax%2B3%2Ay=90000 ALIGN=MIDDLE  ALT=\"5%2Ax%2B3%2Ay=90000\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).<BR>\n" );
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document.write( "  and we can see that the two equations intersect at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15000 ALIGN=MIDDLE  ALT=\"15000\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5000 ALIGN=MIDDLE  ALT=\"5000\"   DISABLED_style= \"max-width:90%; height:auto;\" >). This verifies our answer.</TD></TR>\n" );
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document.write( "note: ignore the graph. It didn't turn out right\r<BR>\n" );
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document.write( "So $15,000 was invested at 5% and $5,000 was invested at 3% </SPAN>\n" );
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