document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=474514>Question 474514</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following Using Cramer's rule\r<BR>\n" );
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document.write( "1.) 8x+5y=7<BR>\n" );
document.write( "    6x-7y=59\r<BR>\n" );
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document.write( "2.) 3x-7y=34<BR>\n" );
document.write( "    8x+9y=-20\r<BR>\n" );
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document.write( "3.) 3x-2y+2z=0<BR>\n" );
document.write( "    4x+3y-z=-15<BR>\n" );
document.write( "    2x+5y+4z=-10 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #325447 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=325447\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "1.) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%2B5y=7 ALIGN=MIDDLE  ALT=\"8x%2B5y=7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+6x-7y=59 ALIGN=MIDDLE  ALT=\"+6x-7y=59\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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=cramers_rule_2x2.solver>Using Cramer's Rule to Solve Systems with 2 variables</A></TD></TR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%288%2Ax%2B5%2Ay=7%2C6%2Ax%2B-7%2Ay=59%29 ALIGN=MIDDLE  ALT=\"system%288%2Ax%2B5%2Ay=7%2C6%2Ax%2B-7%2Ay=59%29\"   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%282%2C2%2C8%2C5%2C6%2C-7%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2C8%2C5%2C6%2C-7%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=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=59 ALIGN=MIDDLE  ALT=\"59\"   DISABLED_style= \"max-width:90%; height:auto;\" > which are highlighted here: <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%288%2Ax%2B5%2Ay=highlight%287%29%2C6%2Ax%2B-7%2Ay=highlight%2859%29%29 ALIGN=MIDDLE  ALT=\"system%288%2Ax%2B5%2Ay=highlight%287%29%2C6%2Ax%2B-7%2Ay=highlight%2859%29%29\"   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=%288%29%28-7%29-%285%29%286%29=-86 ALIGN=MIDDLE  ALT=\"abs%28A%29=%288%29%28-7%29-%285%29%286%29=-86\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Remember that the determinant of the 2x2 matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%29=ad-bc ALIGN=MIDDLE  ALT=\"abs%28A%29=ad-bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If you need help with calculating the determinant of any two by two matrices, then check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-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%282%2C2%2Chighlight%287%29%2C5%2Chighlight%2859%29%2C-7%29%29 ALIGN=MIDDLE  ALT=\"A%5Bx%5D=%28matrix%282%2C2%2Chighlight%287%29%2C5%2Chighlight%2859%29%2C-7%29%29\"   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-2x2-matrix.solver>determinant</a> 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=%287%29%28-7%29-%285%29%2859%29=-344 ALIGN=MIDDLE  ALT=\"abs%28A%5Bx%5D%29=%287%29%28-7%29-%285%29%2859%29=-344\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Once again, remember that the determinant of the 2x2 matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%29=ad-bc ALIGN=MIDDLE  ALT=\"abs%28A%29=ad-bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><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=%28-344%29%2F%28-86%29=4 ALIGN=MIDDLE  ALT=\"x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%28-344%29%2F%28-86%29=4\"   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=4 ALIGN=MIDDLE  ALT=\"x=4\"   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 solution. Let's reset by letting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%282%2C2%2C8%2C5%2C6%2C-7%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2C8%2C5%2C6%2C-7%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%282%2C2%2C8%2Chighlight%287%29%2C6%2Chighlight%2859%29%29%29 ALIGN=MIDDLE  ALT=\"A%5Bx%5D=%28matrix%282%2C2%2C8%2Chighlight%287%29%2C6%2Chighlight%2859%29%29%29\"   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-2x2-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=%288%29%2859%29-%287%29%286%29=430 ALIGN=MIDDLE  ALT=\"abs%28A%5By%5D%29=%288%29%2859%29-%287%29%286%29=430\"   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=%28430%29%2F%28-86%29=-5 ALIGN=MIDDLE  ALT=\"y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%28430%29%2F%28-86%29=-5\"   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=-5 ALIGN=MIDDLE  ALT=\"y=-5\"   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 solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-5 ALIGN=MIDDLE  ALT=\"y=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > giving the ordered pair (4, -5)<BR>\n" );
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document.write( "  Once again, Cramer's Rule is dependent on determinants. Take a look at this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-matrix.solver>2x2 Determinant Solver</a> if you need more practice with determinants.<BR>\n" );
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document.write( "2.) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-7y=34 ALIGN=MIDDLE  ALT=\"3x-7y=34\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%2B9y=-20 ALIGN=MIDDLE  ALT=\"8x%2B9y=-20\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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=cramers_rule_2x2.solver>Using Cramer's Rule to Solve Systems with 2 variables</A></TD></TR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283%2Ax%2B-7%2Ay=34%2C8%2Ax%2B9%2Ay=-20%29 ALIGN=MIDDLE  ALT=\"system%283%2Ax%2B-7%2Ay=34%2C8%2Ax%2B9%2Ay=-20%29\"   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%282%2C2%2C3%2C-7%2C8%2C9%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2C3%2C-7%2C8%2C9%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=34 ALIGN=MIDDLE  ALT=\"34\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-20 ALIGN=MIDDLE  ALT=\"-20\"   DISABLED_style= \"max-width:90%; height:auto;\" > which are highlighted here: <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283%2Ax%2B-7%2Ay=highlight%2834%29%2C8%2Ax%2B9%2Ay=highlight%28-20%29%29 ALIGN=MIDDLE  ALT=\"system%283%2Ax%2B-7%2Ay=highlight%2834%29%2C8%2Ax%2B9%2Ay=highlight%28-20%29%29\"   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=%283%29%289%29-%28-7%29%288%29=83 ALIGN=MIDDLE  ALT=\"abs%28A%29=%283%29%289%29-%28-7%29%288%29=83\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Remember that the determinant of the 2x2 matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%29=ad-bc ALIGN=MIDDLE  ALT=\"abs%28A%29=ad-bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If you need help with calculating the determinant of any two by two matrices, then check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-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( "  Now compute the <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-matrix.solver>determinant</a> 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=%2834%29%289%29-%28-7%29%28-20%29=166 ALIGN=MIDDLE  ALT=\"abs%28A%5Bx%5D%29=%2834%29%289%29-%28-7%29%28-20%29=166\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Once again, remember that the determinant of the 2x2 matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%29=ad-bc ALIGN=MIDDLE  ALT=\"abs%28A%29=ad-bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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=%28166%29%2F%2883%29=2 ALIGN=MIDDLE  ALT=\"x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%28166%29%2F%2883%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So the first solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  ---------------------------------------------------------<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  We'll follow the same basic idea to find the other solution. Let's reset by letting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%282%2C2%2C3%2C-7%2C8%2C9%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2C3%2C-7%2C8%2C9%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > again (this is the coefficient matrix).<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D=%28matrix%282%2C2%2C3%2Chighlight%2834%29%2C8%2Chighlight%28-20%29%29%29 ALIGN=MIDDLE  ALT=\"A%5Bx%5D=%28matrix%282%2C2%2C3%2Chighlight%2834%29%2C8%2Chighlight%28-20%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now compute the <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-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=%283%29%28-20%29-%2834%29%288%29=-332 ALIGN=MIDDLE  ALT=\"abs%28A%5By%5D%29=%283%29%28-20%29-%2834%29%288%29=-332\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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=%28-332%29%2F%2883%29=-4 ALIGN=MIDDLE  ALT=\"y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%28-332%29%2F%2883%29=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  So the second solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-4 ALIGN=MIDDLE  ALT=\"y=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  ====================================================================================<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Final Answer:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So the solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-4 ALIGN=MIDDLE  ALT=\"y=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > giving the ordered pair (2, -4)<BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Once again, Cramer's Rule is dependent on determinants. Take a look at this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-matrix.solver>2x2 Determinant Solver</a> if you need more practice with determinants.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  </TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3.) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-2y%2B2z=0 ALIGN=MIDDLE  ALT=\"3x-2y%2B2z=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%2B3y-z=-15 ALIGN=MIDDLE  ALT=\"4x%2B3y-z=-15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B5y%2B4z=-10 ALIGN=MIDDLE  ALT=\"2x%2B5y%2B4z=-10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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" );
document.write( "<TR><TD  ALIGN=LEFT><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283%2Ax%2B-2%2Ay%2B2%2Az=0%2C4%2Ax%2B3%2Ay%2B-1%2Az=-15%2C2%2Ax%2B5%2Ay%2B4%2Az=-10%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%283%2C3%2C3%2C-2%2C2%2C4%2C3%2C-1%2C2%2C5%2C4%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C3%2C-2%2C2%2C4%2C3%2C-1%2C2%2C5%2C4%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" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-15 ALIGN=MIDDLE  ALT=\"-15\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10 ALIGN=MIDDLE  ALT=\"-10\"   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%283%2Ax%2B-2%2Ay%2B2%2Az=highlight%280%29%2C4%2Ax%2B3%2Ay%2B-1%2Az=highlight%28-15%29%2C2%2Ax%2B5%2Ay%2B4%2Az=highlight%28-10%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <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( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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=115 ALIGN=MIDDLE  ALT=\"abs%28A%29=115\"   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" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  ---------------------------------------------------------<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D=%28matrix%283%2C3%2Chighlight%280%29%2C-2%2C2%2Chighlight%28-15%29%2C3%2C-1%2Chighlight%28-10%29%2C5%2C4%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
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document.write( "  <BR>\n" );
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=-230 ALIGN=MIDDLE  ALT=\"abs%28A%5Bx%5D%29=-230\"   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" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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=%28-230%29%2F%28115%29=-2 ALIGN=MIDDLE  ALT=\"x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%28-230%29%2F%28115%29=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  So the first solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  ---------------------------------------------------------<BR>\n" );
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document.write( "  <BR>\n" );
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%2C3%2C-2%2C2%2C4%2C3%2C-1%2C2%2C5%2C4%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C3%2C-2%2C2%2C4%2C3%2C-1%2C2%2C5%2C4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > again (this is the coefficient matrix).<BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D=%28matrix%283%2C3%2C3%2Chighlight%280%29%2C2%2C4%2Chighlight%28-15%29%2C-1%2C2%2Chighlight%28-10%29%2C4%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <BR>\n" );
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=-230 ALIGN=MIDDLE  ALT=\"abs%28A%5By%5D%29=-230\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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=%28-230%29%2F%28115%29=-2 ALIGN=MIDDLE  ALT=\"y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%28-230%29%2F%28115%29=-2\"   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=-2 ALIGN=MIDDLE  ALT=\"y=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  ---------------------------------------------------------<BR>\n" );
document.write( "  <BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Let's reset again by letting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%283%2C3%2C3%2C-2%2C2%2C4%2C3%2C-1%2C2%2C5%2C4%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C3%2C-2%2C2%2C4%2C3%2C-1%2C2%2C5%2C4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is the coefficient matrix.<BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bz%5D=%28matrix%283%2C3%2C3%2C-2%2Chighlight%280%29%2C4%2C3%2Chighlight%28-15%29%2C2%2C5%2Chighlight%28-10%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <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=115 ALIGN=MIDDLE  ALT=\"abs%28A%5Bz%5D%29=115\"   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=%28115%29%2F%28115%29=1 ALIGN=MIDDLE  ALT=\"z=%28abs%28A%5Bz%5D%29%29%2F%28abs%28A%29%29=%28115%29%2F%28115%29=1\"   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=1 ALIGN=MIDDLE  ALT=\"z=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  ====================================================================================<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=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2 ALIGN=MIDDLE  ALT=\"y=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=1 ALIGN=MIDDLE  ALT=\"z=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > giving the ordered triple (-2, -2, 1)<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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