document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=114878>Question 114878</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need a example of how to use the Cramer Rule step by step. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #84786 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=84786\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hard to tell where to start with this one.  I'll just assume that you know how to evaluate a determinant -- if that isn't true, write back and we'll take a step back.\r<BR>\n" );
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document.write( "Let's say you have a system of three equations:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5Dx%2Bb%5B1%5Dy%2Bc%5B1%5Dz=d%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5Dx%2Bb%5B1%5Dy%2Bc%5B1%5Dz=d%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5Dx%2Bb%5B2%5Dy%2Bc%5B2%5Dz=d%5B2%5D ALIGN=MIDDLE  ALT=\"a%5B2%5Dx%2Bb%5B2%5Dy%2Bc%5B2%5Dz=d%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5Dx%2Bb%5B3%5Dy%2Bc%5B3%5Dz=d%5B3%5D ALIGN=MIDDLE  ALT=\"a%5B3%5Dx%2Bb%5B3%5Dy%2Bc%5B3%5Dz=d%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "First, create the coefficient Determinant,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=%28matrix%283%2C3%2C%0D%0Aa%5B1%5D%2Cb%5B1%5D%2Cc%5B1%5D%2C%0D%0Aa%5B2%5D%2Cb%5B2%5D%2Cc%5B2%5D%2C%0D%0Aa%5B3%5D%2Cb%5B3%5D%2Cc%5B1%5D%0D%0A%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, and the constant matrix<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C1%2Cgreen%28d%5B1%5D%29%2Cgreen%28d%5B2%5D%29%2Cgreen%28d%5B3%5D%29%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C1%2Cgreen%28d%5B1%5D%29%2Cgreen%28d%5B2%5D%29%2Cgreen%28d%5B3%5D%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now, replace the first column in the coefficient Determinant with the values in the constant matrix to get the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5Bx%5D ALIGN=MIDDLE  ALT=\"D%5Bx%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > determinant,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5Bred%28x%29%5D=%28matrix%283%2C3%2C%0D%0Ahighlight%28green%28d%5B1%5D%29%29%2Cb%5B1%5D%2Cc%5B1%5D%2C%0D%0Ahighlight%28green%28d%5B2%5D%29%29%2Cb%5B2%5D%2Cc%5B2%5D%2C%0D%0Ahighlight%28green%28d%5B3%5D%29%29%2Cb%5B3%5D%2Cc%5B1%5D%0D%0A%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Do this two more times, replacing the second and third columns with the constant matrix values to get the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5By%5D ALIGN=MIDDLE  ALT=\"D%5By%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5Bz%5D ALIGN=MIDDLE  ALT=\"D%5Bz%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > determinants.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5Bgreen%28y%29%5D=%28matrix%283%2C3%2C%0D%0Aa%5B1%5D%2Chighlight%28green%28d%5B1%5D%29%29%2Cc%5B1%5D%2C%0D%0Aa%5B2%5D%2Chighlight%28green%28d%5B2%5D%29%29%2Cc%5B2%5D%2C%0D%0Aa%5B3%5D%2Chighlight%28green%28d%5B3%5D%29%29%2Cc%5B1%5D%0D%0A%29%29 ALIGN=MIDDLE   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=D%5Bblue%28z%29%5D=%28matrix%283%2C3%2C%0D%0Aa%5B1%5D%2Cb%5B1%5D%2Chighlight%28green%28d%5B1%5D%29%29%2C%0D%0Aa%5B2%5D%2Cb%5B2%5D%2Chighlight%28green%28d%5B2%5D%29%29%2C%0D%0Aa%5B3%5D%2Cb%5B3%5D%2Chighlight%28green%28d%5B1%5D%29%29%0D%0A%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now, evaluate all four determinants, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5Bred%28x%29%5D ALIGN=MIDDLE  ALT=\"D%5Bred%28x%29%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5Bgreen%28y%29%5D ALIGN=MIDDLE  ALT=\"D%5Bgreen%28y%29%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5Bblue%28z%29%5D ALIGN=MIDDLE  ALT=\"D%5Bblue%28z%29%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (I've already assumed that you know how to do this, so I'm continuing with Cramer's rule.)\r<BR>\n" );
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document.write( "Cramer's Rule says:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=D%5Bred%28x%29%5D%2FD ALIGN=MIDDLE  ALT=\"x=D%5Bred%28x%29%5D%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=D%5Bgreen%28y%29%5D%2FD ALIGN=MIDDLE  ALT=\"y=D%5Bgreen%28y%29%5D%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=D%5Bblue%28z%29%5D%2FD ALIGN=MIDDLE  ALT=\"z=D%5Bblue%28z%29%5D%2FD\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "All you have to do is the arithmetic.\r<BR>\n" );
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document.write( "Hope this helps.\r<BR>\n" );
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document.write( "John\r<BR>\n" );
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document.write( "P.S.  I forgot to mention one very important detail, if the coefficient determinant is zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=0 ALIGN=MIDDLE  ALT=\"D=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then Cramer's Rule doesn't work because you can't divide by zero.  The problem is you may still have to figure out if you have an inconsistent system (parallel or possibly skew lines with no solution) or a consistent and undetermined system (coincident lines) with an infinite number of solutions.  Cramer's rule cannot answer this question, and you will have to find some other solution method. </SPAN>\n" );
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