document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=385407>Question 385407</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.\r<BR>\n" );
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document.write( "  x - y + 2z = 6<BR>\n" );
document.write( "4x + z = 1<BR>\n" );
document.write( "  x + 4y + z = -15 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #272724 by </B><A HREF=/tutors/aboutme.mpl?userid=Jk22><B>Jk22(389)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Jk22><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=Jk22><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=272724\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <TABLE  BORDER=1>\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=determinant_of_3x3_matrix.solver>Finding the Determinant of a 3x3 Matrix</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT><br>If you have the general 3x3 matrix:<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>the <a  rel=nofollow HREF=\"http://en.wikipedia.org/wiki/Determinant#3-by-3_matrices\">determinant</a> is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%2Aabs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29-b%2Aabs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29%2Bc%2Aabs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>Which further breaks down to:<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%28ei-fh%29-b%2A%28di-fg%29%2Bc%28dh-eg%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>Note: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are determinants themselves.<br> If you need help finding the determinant of 2x2 matrices (which is <i>required</i> to find the determinant of 3x3 matrices), check out this <a  rel=nofollow HREF=\"http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-matrix.solver\">solver</a><br><br>--------------------------------------------------------------<br><br><br>From the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-1 ALIGN=MIDDLE  ALT=\"b=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=2 ALIGN=MIDDLE  ALT=\"c=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=4 ALIGN=MIDDLE  ALT=\"d=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e=0 ALIGN=MIDDLE  ALT=\"e=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=1 ALIGN=MIDDLE  ALT=\"f=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g=1 ALIGN=MIDDLE  ALT=\"g=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=4 ALIGN=MIDDLE  ALT=\"h=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i=1 ALIGN=MIDDLE  ALT=\"i=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%28ei-fh%29-b%2A%28di-fg%29%2Bc%28dh-eg%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the general 3x3 determinant.<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29=%281%29%28%280%29%281%29-%281%29%284%29%29-%28-1%29%28%284%29%281%29-%281%29%281%29%29%2B%282%29%28%284%29%284%29-%280%29%281%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the given values (see above)<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29=%281%29%280-4%29-%28-1%29%284-1%29%2B%282%29%2816-0%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29=%281%29%28-4%29-%28-1%29%283%29%2B%282%29%2816%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29=-4--3%2B32 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29=-4--3%2B32\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29=31 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29=31\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.<br><br><br>======================================================================<br><br><br>Answer:<br><br>So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29=31 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29=31\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  which means that the determinant of the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font color=\"red\">31</font><br></TD></TR>\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%281%2Ax%2B-1%2Ay%2B2%2Az=6%2C4%2Ax%2B0%2Ay%2B1%2Az=1%2C1%2Ax%2B4%2Ay%2B1%2Az=-15%29 ALIGN=MIDDLE  ALT=\"system%281%2Ax%2B-1%2Ay%2B2%2Az=6%2C4%2Ax%2B0%2Ay%2B1%2Az=1%2C1%2Ax%2B4%2Ay%2B1%2Az=-15%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%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%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=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <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 they are highlighted here: <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%281%2Ax%2B-1%2Ay%2B2%2Az=highlight%286%29%2C4%2Ax%2B0%2Ay%2B1%2Az=highlight%281%29%2C1%2Ax%2B4%2Ay%2B1%2Az=highlight%28-15%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=31 ALIGN=MIDDLE  ALT=\"abs%28A%29=31\"   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%286%29%2C-1%2C2%2Chighlight%281%29%2C0%2C1%2Chighlight%28-15%29%2C4%2C1%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=0 ALIGN=MIDDLE  ALT=\"abs%28A%5Bx%5D%29=0\"   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=%280%29%2F%2831%29=0 ALIGN=MIDDLE  ALT=\"x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%280%29%2F%2831%29=0\"   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=0 ALIGN=MIDDLE  ALT=\"x=0\"   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%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%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%2C1%2Chighlight%286%29%2C2%2C4%2Chighlight%281%29%2C1%2C1%2Chighlight%28-15%29%2C1%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=-124 ALIGN=MIDDLE  ALT=\"abs%28A%5By%5D%29=-124\"   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=%28-124%29%2F%2831%29=-4 ALIGN=MIDDLE  ALT=\"y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%28-124%29%2F%2831%29=-4\"   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=-4 ALIGN=MIDDLE  ALT=\"y=-4\"   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%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C1%2C-1%2C2%2C4%2C0%2C1%2C1%2C4%2C1%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%2C1%2C-1%2Chighlight%286%29%2C4%2C0%2Chighlight%281%29%2C1%2C4%2Chighlight%28-15%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=31 ALIGN=MIDDLE  ALT=\"abs%28A%5Bz%5D%29=31\"   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=%2831%29%2F%2831%29=1 ALIGN=MIDDLE  ALT=\"z=%28abs%28A%5Bz%5D%29%29%2F%28abs%28A%29%29=%2831%29%2F%2831%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=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <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;\" >, 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 (0, -4, 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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