document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=868933>Question 868933</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1] . Using matrix method, solve the following system of linear equations :<BR>\n" );
document.write( "2x – y = 4, 2y + z = 5, z + 2x = 7. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #523957 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=523957\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 2 –1 +0 4<BR>\n" );
document.write( "0 +2 +1 5<BR>\n" );
document.write( "2 +0 +1 7\r<BR>\n" );
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document.write( "gauss jordan method<BR>\n" );
document.write( "Divide row1 by 2<BR>\n" );
document.write( "1	 -1/2	 0	 2<BR>\n" );
document.write( "0	 2	 1	 5<BR>\n" );
document.write( "2	 0	 1	 7\r<BR>\n" );
document.write( "\n" );
document.write( "Add (-2 * row1) to row3<BR>\n" );
document.write( "1	 -1/2	 0	 2<BR>\n" );
document.write( "0	 2	 1	 5<BR>\n" );
document.write( "0	 1	 1	 3\r<BR>\n" );
document.write( "\n" );
document.write( "Divide row2 by 2<BR>\n" );
document.write( "1	 -1/2	 0	 2<BR>\n" );
document.write( "0	 1	 1/2	 5/2<BR>\n" );
document.write( "0	 1	 1	 3\r<BR>\n" );
document.write( "\n" );
document.write( "Add (-1 * row2) to row3<BR>\n" );
document.write( "1	 -1/2	 0	 2<BR>\n" );
document.write( "0	 1	 1/2	 5/2<BR>\n" );
document.write( "0	 0	 1/2	 1/2\r<BR>\n" );
document.write( "\n" );
document.write( "Divide row3 by 1/2<BR>\n" );
document.write( "1	 -1/2	 0	 2<BR>\n" );
document.write( "0	 1	 1/2	 5/2<BR>\n" );
document.write( "0	 0	 1	 1\r<BR>\n" );
document.write( "\n" );
document.write( "Add (-1/2 * row3) to row2<BR>\n" );
document.write( "1	 -1/2	 0	 2<BR>\n" );
document.write( "0	 1	 0	 2<BR>\n" );
document.write( "0	 0	 1	 1\r<BR>\n" );
document.write( "\n" );
document.write( "Add (1/2 * row2) to row1<BR>\n" );
document.write( "1	 0	 0	 3<BR>\n" );
document.write( "0	 1	 0	 2<BR>\n" );
document.write( "0	 0	 1	 1<BR>\n" );
document.write( "x=3 y=2 z=1<BR>\n" );
document.write( "Cramers rule<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%282%2Ax%2B-1%2Ay%2B0%2Az=4%2C0%2Ax%2B2%2Ay%2B1%2Az=5%2C2%2Ax%2B0%2Ay%2B1%2Az=7%29 ALIGN=MIDDLE  ALT=\"system%282%2Ax%2B-1%2Ay%2B0%2Az=4%2C0%2Ax%2B2%2Ay%2B1%2Az=5%2C2%2Ax%2B0%2Ay%2B1%2Az=7%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%2C2%2C-1%2C0%2C0%2C2%2C1%2C2%2C0%2C1%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C2%2C-1%2C0%2C0%2C2%2C1%2C2%2C0%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=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <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 they are highlighted here: <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282%2Ax%2B-1%2Ay%2B0%2Az=highlight%284%29%2C0%2Ax%2B2%2Ay%2B1%2Az=highlight%285%29%2C2%2Ax%2B0%2Ay%2B1%2Az=highlight%287%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=2 ALIGN=MIDDLE  ALT=\"abs%28A%29=2\"   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%284%29%2C-1%2C0%2Chighlight%285%29%2C2%2C1%2Chighlight%287%29%2C0%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=6 ALIGN=MIDDLE  ALT=\"abs%28A%5Bx%5D%29=6\"   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=%286%29%2F%282%29=3 ALIGN=MIDDLE  ALT=\"x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%286%29%2F%282%29=3\"   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=3 ALIGN=MIDDLE  ALT=\"x=3\"   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%2C2%2C-1%2C0%2C0%2C2%2C1%2C2%2C0%2C1%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C2%2C-1%2C0%2C0%2C2%2C1%2C2%2C0%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%2C2%2Chighlight%284%29%2C0%2C0%2Chighlight%285%29%2C1%2C2%2Chighlight%287%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=4 ALIGN=MIDDLE  ALT=\"abs%28A%5By%5D%29=4\"   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=%284%29%2F%282%29=2 ALIGN=MIDDLE  ALT=\"y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%284%29%2F%282%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( "  Let's reset again by letting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%283%2C3%2C2%2C-1%2C0%2C0%2C2%2C1%2C2%2C0%2C1%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C2%2C-1%2C0%2C0%2C2%2C1%2C2%2C0%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%2C2%2C-1%2Chighlight%284%29%2C0%2C2%2Chighlight%285%29%2C2%2C0%2Chighlight%287%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=2 ALIGN=MIDDLE  ALT=\"abs%28A%5Bz%5D%29=2\"   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=%282%29%2F%282%29=1 ALIGN=MIDDLE  ALT=\"z=%28abs%28A%5Bz%5D%29%29%2F%28abs%28A%29%29=%282%29%2F%282%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=3 ALIGN=MIDDLE  ALT=\"x=3\"   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 (3, 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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