document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1099909>Question 1099909</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Jack inherited 250,000 pesos and invested money in SM, Meralco, and Manila Water. After a year, he got a small return of 16,200 pesos from the three investments. SM returned 6%, Meralco returned 7%, and Manila Water returned 8%. There was 60,000 more invested in Meralco than in Manila Water. How much did he invest in SM, Meralco, and Manila Water? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #714413 by </B><A HREF=/tutors/aboutme.mpl?userid=richwmiller><B>richwmiller(17219)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richwmiller><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=richwmiller><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=714413\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .06x+.07y+.08z=16200<BR>\n" );
document.write( "x+y+z=250000<BR>\n" );
document.write( "0x+y-z=60000\r<BR>\n" );
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document.write( ".06,.07,.08,16200<BR>\n" );
document.write( "1,1,1,250000<BR>\n" );
document.write( "0,1,-1,60000\r<BR>\n" );
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document.write( "SM=150000, Meralco=80000, Manila Water=20000<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=cramers_rule_3x3.solver>Using Cramer's Rule to Solve Systems with 3 variables</A></TD></TR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%280.06%2Ax%2B0.07%2Ay%2B0.08%2Az=16200%2C1%2Ax%2B1%2Ay%2B1%2Az=250000%2C0%2Ax%2B1%2Ay%2B-1%2Az=60000%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  First let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%283%2C3%2C0.06%2C0.07%2C0.08%2C1%2C1%2C1%2C0%2C1%2C-1%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C0.06%2C0.07%2C0.08%2C1%2C1%2C1%2C0%2C1%2C-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This is the matrix formed by the coefficients of the given system of equations.<BR>\n" );
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document.write( "  Take note that the right hand values of the system are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16200 ALIGN=MIDDLE  ALT=\"16200\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=250000 ALIGN=MIDDLE  ALT=\"250000\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60000 ALIGN=MIDDLE  ALT=\"60000\"   DISABLED_style= \"max-width:90%; height:auto;\" > and they are highlighted here: <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%280.06%2Ax%2B0.07%2Ay%2B0.08%2Az=highlight%2816200%29%2C1%2Ax%2B1%2Ay%2B1%2Az=highlight%28250000%29%2C0%2Ax%2B1%2Ay%2B-1%2Az=highlight%2860000%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  These values are important as they will be used to replace the columns of the matrix A.<BR>\n" );
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document.write( "  Now let's calculate the the <a  rel=nofollow HREF=http://en.wikipedia.org/wiki/Determinant>determinant</a> of the matrix A to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%29=0.03 ALIGN=MIDDLE  ALT=\"abs%28A%29=0.03\"   DISABLED_style= \"max-width:90%; height:auto;\" >. To save space, I'm not showing the calculations for the determinant. However, if you need help with calculating the determinant of the matrix A, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-3x3-matrix.solver>solver</a>.<BR>\n" );
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document.write( "  Notation note: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%29 ALIGN=MIDDLE  ALT=\"abs%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > denotes the determinant of the matrix A.<BR>\n" );
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document.write( "  Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D ALIGN=MIDDLE  ALT=\"A%5Bx%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since we're replacing the 'x' column so to speak).<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D=%28matrix%283%2C3%2Chighlight%2816200%29%2C0.07%2C0.08%2Chighlight%28250000%29%2C1%2C1%2Chighlight%2860000%29%2C1%2C-1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now compute the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D ALIGN=MIDDLE  ALT=\"A%5Bx%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%5Bx%5D%29=4500 ALIGN=MIDDLE  ALT=\"abs%28A%5Bx%5D%29=4500\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Again, as a space saver, I didn't include the calculations of the determinant. Check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-3x3-matrix.solver>solver</a> to see how to find this determinant.<BR>\n" );
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document.write( "  To find the first solution, simply divide the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D ALIGN=MIDDLE  ALT=\"A%5Bx%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%284500%29%2F%280.03%29=150000 ALIGN=MIDDLE  ALT=\"x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%284500%29%2F%280.03%29=150000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So the first solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=150000 ALIGN=MIDDLE  ALT=\"x=150000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  We'll follow the same basic idea to find the other two solutions. Let's reset by letting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%283%2C3%2C0.06%2C0.07%2C0.08%2C1%2C1%2C1%2C0%2C1%2C-1%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C0.06%2C0.07%2C0.08%2C1%2C1%2C1%2C0%2C1%2C-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > again (this is the coefficient matrix).<BR>\n" );
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document.write( "  Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5By%5D ALIGN=MIDDLE  ALT=\"A%5By%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since we're replacing the 'y' column in a way).<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D=%28matrix%283%2C3%2C0.06%2Chighlight%2816200%29%2C0.08%2C1%2Chighlight%28250000%29%2C1%2C0%2Chighlight%2860000%29%2C-1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now compute the <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-3x3-matrix.solver>determinant</a> of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5By%5D ALIGN=MIDDLE  ALT=\"A%5By%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%5By%5D%29=2400 ALIGN=MIDDLE  ALT=\"abs%28A%5By%5D%29=2400\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "  To find the second solution, divide the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5By%5D ALIGN=MIDDLE  ALT=\"A%5By%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%282400%29%2F%280.03%29=80000 ALIGN=MIDDLE  ALT=\"y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%282400%29%2F%280.03%29=80000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So the second solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=80000 ALIGN=MIDDLE  ALT=\"y=80000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Let's reset again by letting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%283%2C3%2C0.06%2C0.07%2C0.08%2C1%2C1%2C1%2C0%2C1%2C-1%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C0.06%2C0.07%2C0.08%2C1%2C1%2C1%2C0%2C1%2C-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is the coefficient matrix.<BR>\n" );
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document.write( "  Replace the third column of A (that corresponds to the variable 'z') with the values that form the right hand side of the system of equations. We will denote this new matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bz%5D ALIGN=MIDDLE  ALT=\"A%5Bz%5D\"   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=A%5Bz%5D=%28matrix%283%2C3%2C0.06%2C0.07%2Chighlight%2816200%29%2C1%2C1%2Chighlight%28250000%29%2C0%2C1%2Chighlight%2860000%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now compute the <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-3x3-matrix.solver>determinant</a> of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bz%5D ALIGN=MIDDLE  ALT=\"A%5Bz%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%5Bz%5D%29=600 ALIGN=MIDDLE  ALT=\"abs%28A%5Bz%5D%29=600\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "  To find the third solution, divide the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bz%5D ALIGN=MIDDLE  ALT=\"A%5Bz%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=%28abs%28A%5Bz%5D%29%29%2F%28abs%28A%29%29=%28600%29%2F%280.03%29=20000 ALIGN=MIDDLE  ALT=\"z=%28abs%28A%5Bz%5D%29%29%2F%28abs%28A%29%29=%28600%29%2F%280.03%29=20000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So the third solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=20000 ALIGN=MIDDLE  ALT=\"z=20000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Final Answer:<BR>\n" );
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document.write( "  So the three solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=150000 ALIGN=MIDDLE  ALT=\"x=150000\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=80000 ALIGN=MIDDLE  ALT=\"y=80000\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=20000 ALIGN=MIDDLE  ALT=\"z=20000\"   DISABLED_style= \"max-width:90%; height:auto;\" > giving the ordered triple (150000, 80000, 20000)<BR>\n" );
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document.write( "  Note: there is a lot of work that is hidden in finding the determinants. Take a look at this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-3x3-matrix.solver>3x3 Determinant Solver</a> to see how to get each determinant.<BR>\n" );
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document.write( "gauss jordan <BR>\n" );
document.write( "same as elimination without the variables<BR>\n" );
document.write( ".06x+.07y+.08z=16200<BR>\n" );
document.write( "x+y+z=250000<BR>\n" );
document.write( "0x+y-z=60000<BR>\n" );
document.write( "original 3*4 matrix<BR>\n" );
document.write( "0.06,0.07,0.08,16200<BR>\n" );
document.write( "1,1,1,250000<BR>\n" );
document.write( "0,1,-1,60000\r<BR>\n" );
document.write( "\n" );
document.write( "divide row 1 by 0.06<BR>\n" );
document.write( "1,1.16666667,1.33333333,270000<BR>\n" );
document.write( "1,1,1,250000<BR>\n" );
document.write( "0,1,-1,60000\r<BR>\n" );
document.write( "\n" );
document.write( "add  -1*row 1 to row 2<BR>\n" );
document.write( "1,1.16666667,1.33333333,270000<BR>\n" );
document.write( "0,-0.16666667,-0.33333333,-20000<BR>\n" );
document.write( "0,1,-1,60000\r<BR>\n" );
document.write( "\n" );
document.write( "add  0*row 1 to row 3<BR>\n" );
document.write( "1,1.16666667,1.33333333,270000<BR>\n" );
document.write( "0,-0.16666667,-0.33333333,-20000<BR>\n" );
document.write( "0,1,-1,60000\r<BR>\n" );
document.write( "\n" );
document.write( "divide row 2 by -0.16666667<BR>\n" );
document.write( "1,1.16666667,1.33333333,270000<BR>\n" );
document.write( "0,1,2.0,120000.0<BR>\n" );
document.write( "0,1,-1,60000\r<BR>\n" );
document.write( "\n" );
document.write( "add  -1*row 2 to row 3<BR>\n" );
document.write( "1,1.16666667,1.33333333,270000<BR>\n" );
document.write( "0,1,2.0,120000.0<BR>\n" );
document.write( "0,0,-3.0,-60000.0\r<BR>\n" );
document.write( "\n" );
document.write( "divide row 3 by -3.0<BR>\n" );
document.write( "1,1.16666667,1.33333333,270000<BR>\n" );
document.write( "0,1,2.0,120000.0<BR>\n" );
document.write( "0,0,1,20000.0\r<BR>\n" );
document.write( "\n" );
document.write( "add  -2.0*row 3 to row 2<BR>\n" );
document.write( "1,1.16666667,1.33333333,270000<BR>\n" );
document.write( "0,1,0,80000.0<BR>\n" );
document.write( "0,0,1,20000.0\r<BR>\n" );
document.write( "\n" );
document.write( "add  -1.33333333*row 3 to row 1<BR>\n" );
document.write( "1,1.16666667,0,243333.333<BR>\n" );
document.write( "0,1,0,80000.0<BR>\n" );
document.write( "0,0,1,20000.0\r<BR>\n" );
document.write( "\n" );
document.write( "add  -1.16666667*row 2 to row 1<BR>\n" );
document.write( "1,0,0,150000.0<BR>\n" );
document.write( "0,1,0,80000.0<BR>\n" );
document.write( "0,0,1,20000.0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1 150000.0<BR>\n" );
document.write( "2 80000.0<BR>\n" );
document.write( "3 20000.0<BR>\n" );
document.write( "done<BR>\n" );
document.write( "check<BR>\n" );
document.write( ".06*150000+.07*80000+.08*20000=16200<BR>\n" );
document.write( "9000+5600+1600=16200<BR>\n" );
document.write( "16200=16200<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );