document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=53677>Question 53677</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This is a question involving 3 variables, solve by elimination:\r<BR>\n" );
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document.write( "2x - 5y + 3z = -1<BR>\n" );
document.write( "x  + 4y - 2z = 9<BR>\n" );
document.write( "x  - 2y - 4z = -5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #35914 by </B><A HREF=/tutors/aboutme.mpl?userid=funmath><B>funmath(2933)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=funmath><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=35914\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You'll like these much better once you are introduced to matrices.  Until then, we eliminate.<BR>\n" );
document.write( "2x-5y+3z=-1<BR>\n" );
document.write( "x+4y-2z=9<BR>\n" );
document.write( "x-2z-4z=-5<BR>\n" );
document.write( "-----------------------------------------------<BR>\n" );
document.write( "(Step 1)Eliminate the x in the first 2 equations:<BR>\n" );
document.write( "2x-5y+3z=-1<BR>\n" );
document.write( "x+4y-2z=9<BR>\n" );
document.write( "----<BR>\n" );
document.write( "Multiply the second equation by -2 and add the two equations together.<BR>\n" );
document.write( "2x-5y+3y=-1<BR>\n" );
document.write( "-2(x+4y-2z)=-2(9)<BR>\n" );
document.write( "---<BR>\n" );
document.write( "2x-5y+3y=-1<BR>\n" );
document.write( "-2x-8y+4z=-18<BR>\n" );
document.write( "_____________<BR>\n" );
document.write( "0x-13y+7z=-19<BR>\n" );
document.write( "The result of step 1 is -13y+7y=-19<BR>\n" );
document.write( "--------------------------------------------------<BR>\n" );
document.write( "(Step 2)Eliminate the x in the second two equations:<BR>\n" );
document.write( "x+4y-2z=9<BR>\n" );
document.write( "x-2y-4z=-5<BR>\n" );
document.write( "----<BR>\n" );
document.write( "Multiply the the last equation by -1 and add the two equations together.<BR>\n" );
document.write( "x+4y-2z=9<BR>\n" );
document.write( "-1(x-2y-4z)=-1(-5)<BR>\n" );
document.write( "---<BR>\n" );
document.write( "x+4y-2z=9<BR>\n" );
document.write( "-x+2y+4z=5<BR>\n" );
document.write( "__________<BR>\n" );
document.write( "0x+6y+2z=14<BR>\n" );
document.write( "The result of step 2 is 6y+2z=14<BR>\n" );
document.write( "-----------------------------------------------------<BR>\n" );
document.write( "(Step 3) Eliminate the z's in the two equations that are the results of steps 1 and 2 and solve for y.<BR>\n" );
document.write( "-13y+7z=-19<BR>\n" );
document.write( "6y+2z=14<BR>\n" );
document.write( "----Multiply the result of step one by-2 and multiply the result of step 2 by 7 and add the two equations together.<BR>\n" );
document.write( "-2(-13z+7z)=-2(-19)<BR>\n" );
document.write( "7(6y+2z)=7(14)<BR>\n" );
document.write( "----<BR>\n" );
document.write( "26y-14z=38<BR>\n" );
document.write( "42y+14z=98<BR>\n" );
document.write( "__________<BR>\n" );
document.write( "68y+0z=136<BR>\n" );
document.write( "68y=136<BR>\n" );
document.write( "68y/68=136/68<BR>\n" );
document.write( "y=2<BR>\n" );
document.write( "----------------------------------------------------------------------<BR>\n" );
document.write( "(Step 4) Substitute y=2 into the result of step 2 and solve for z:<BR>\n" );
document.write( "6(2)+2z=14<BR>\n" );
document.write( "12+2z=14<BR>\n" );
document.write( "-12+12+2z=14-12<BR>\n" );
document.write( "2z=2<BR>\n" );
document.write( "2z/2=2/2<BR>\n" );
document.write( "z=1<BR>\n" );
document.write( "----------------------------------------------------------------------<BR>\n" );
document.write( "(step 5)  Substitute the y=2 that we got in step 3 and the z=1 that we got from step 4 into one of the original equations (I'm picking the last one because it looks easiest) and solve for x.<BR>\n" );
document.write( "x-2(2)-4(1)=-5<BR>\n" );
document.write( "x-4-4=-5<BR>\n" );
document.write( "x-8=-5<BR>\n" );
document.write( "x-8+8=-5+8<BR>\n" );
document.write( "x=3<BR>\n" );
document.write( "-------------------------------------------------------------------------<BR>\n" );
document.write( "(Step 6) Check your answer (3,2,1) in all three equations to see if they check.  There are way too many ways to mess these things up!<BR>\n" );
document.write( "2(3)-5(2)+3(1)=-1<BR>\n" );
document.write( "6-10+3=-1<BR>\n" );
document.write( "-1=-1  Equation 1 checks, we're looking good.<BR>\n" );
document.write( "(3)+4(2)-2(1)=9<BR>\n" );
document.write( "3+8-2=9<BR>\n" );
document.write( "9=9  Equation 2 checks, we're looking better.<BR>\n" );
document.write( "(3)-2(2)-4(1)=-5<BR>\n" );
document.write( "3-4-4=-5  Equation 3 checks, we're right!!!! <BR>\n" );
document.write( "The answer is: (x,y,z)=(3,2,1).<BR>\n" );
document.write( "Happy Calculating!!!!  <BR>\n" );
document.write( "  </SPAN>\n" );
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