document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1006275>Question 1006275</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the system of equations by converting the matrix in reduced echelon form.<BR>\n" );
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document.write( "2x-2y+2z=1<BR>\n" );
document.write( "   3y-z =5<BR>\n" );
document.write( "-2x+3y-z=1\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #622455 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=622455\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This system<BR>\n" );
document.write( "2x-2y+2z=1<BR>\n" );
document.write( "3y-z =5<BR>\n" );
document.write( "-2x+3y-z=1 \r<BR>\n" );
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document.write( "Is the same as this system<BR>\n" );
document.write( "2x-2y+2z=1<BR>\n" );
document.write( "0x+3y-1z=5<BR>\n" );
document.write( "-2x+3y-1z=1 \r<BR>\n" );
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document.write( "Convert to matrix form<BR>\n" );
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document.write( "Add row 1 to row 3. The result will replace row 3<BR>\n" );
document.write( "R1+R3 --> R3<BR>\n" );
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document.write( "Swap row 2 and row 3<BR>\n" );
document.write( "R2 <--> R3<BR>\n" );
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document.write( "Subtract 3 times row 3 from row 2. This result will replace row 3<BR>\n" );
document.write( "R3 - 3*R2 ---> R3<BR>\n" );
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document.write( "Multiply every term of row 3 by -0.25<BR>\n" );
document.write( "-0.25*R3 ---> R3<BR>\n" );
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document.write( "Subtract 1*Row 3 from row 2. Then replace all of row 2 with the result<BR>\n" );
document.write( "R2 - 1*R3 --> R2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left[\begin{array}{ccc|c}2 & -2 & 2 & 1\\0 & 1 & 0 & 1.75\\0 & 0 & 1 & 0.25\end{array}\right]\">\r<BR>\n" );
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document.write( "Subtract 2*Row 3 from row 1. Then replace all of row 1 with the result<BR>\n" );
document.write( "R1 - 2*R3 --> R1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left[\begin{array}{ccc|c}2 & -2 & 0 & 0.5\\0 & 1 & 0 & 1.75\\0 & 0 & 1 & 0.25\end{array}\right]\">\r<BR>\n" );
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document.write( "Subtract -2*Row 2 from row 1. Then replace all of row 1 with the result<BR>\n" );
document.write( "R1 - (-2)*R2 --> R1<BR>\n" );
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document.write( "Multiply all of row 1 by 0.5<BR>\n" );
document.write( "0.5*R1 ---> R1<BR>\n" );
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document.write( "The last matrix above is in reduced row echelon form. So we're done. \r<BR>\n" );
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document.write( "The last thing to do is simply read off the values in the last column. They are: 2, 1.75, 0.25\r<BR>\n" );
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document.write( "So <BR>\n" );
document.write( "x = 2<BR>\n" );
document.write( "y = 1.75<BR>\n" );
document.write( "z = 0.25\r<BR>\n" );
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document.write( "In fraction form<BR>\n" );
document.write( "x = 2<BR>\n" );
document.write( "y = 7/4<BR>\n" );
document.write( "z = 1/4\r<BR>\n" );
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document.write( "The solution written as an ordered triple is (x,y,z) = (2,1.75,0.25)<BR>\n" );
document.write( "In fraction form, the solution as an ordered triple is (x,y,z) = (2,7/4,1/4)\r<BR>\n" );
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document.write( "Check:\r<BR>\n" );
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document.write( "Let's check the answer (x,y,z) = (2,1.75,0.25). To do this, we replace the x,y,z with the values we found above (2, 1.75 and 0.25 respectively) into each equation. The x,y,z values must make ALL of the equations of the system true. \r<BR>\n" );
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document.write( "Let's check equation 1<BR>\n" );
document.write( "2x-2y+2z=1<BR>\n" );
document.write( "2*2-2*1.75+2*0.25=1 ... plug in x = 2,y = 1.75,z = 0.25<BR>\n" );
document.write( "1=1 ... true\r<BR>\n" );
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document.write( "Now let's check equation 2<BR>\n" );
document.write( "3y-z =5<BR>\n" );
document.write( "3*1.75-0.25 =5 ... plug in x = 2,y = 1.75,z = 0.25<BR>\n" );
document.write( "5=5 ... true\r<BR>\n" );
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document.write( "and finally equation 3<BR>\n" );
document.write( "-2x+3y-z=1 <BR>\n" );
document.write( "-2*2+3*1.75-0.25=1  ... plug in x = 2,y = 1.75,z = 0.25<BR>\n" );
document.write( "1=1 ... true\r<BR>\n" );
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document.write( "All three equations are true when (x,y,z) = (2,1.75,0.25). So the solution is confirmed.  </SPAN>\n" );
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