document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=202802>Question 202802</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve for X,Y and Z<BR>\n" );
document.write( "X+10Y+Z=52<BR>\n" );
document.write( " 5X+Y+4Z=15<BR>\n" );
document.write( " X+2y-3Z=12<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 #153098 by </B><A HREF=/tutors/aboutme.mpl?userid=PRMath><B>PRMath(133)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=PRMath><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=153098\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Solve for X,Y and Z<BR>\n" );
document.write( "X+10Y+Z=52<BR>\n" );
document.write( "5X+Y+4Z=15<BR>\n" );
document.write( "X+2y-3Z=12\r<BR>\n" );
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document.write( "First let's choose ANY two equations and eliminate ONE variable.  I'd suggest the first and 3rd equations..........\r<BR>\n" );
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document.write( "X+10Y+Z=52<BR>\n" );
document.write( "X+2y-3Z=12  (Hm... what if we multiplied this equation by -1?)\r<BR>\n" );
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document.write( "X+10Y+Z=52<BR>\n" );
document.write( "-X-2y+3Z=-12  (Do you see how we multiplied this whole equation by -1?)<BR>\n" );
document.write( "_____________  Now let's add these equations<BR>\n" );
document.write( "8y + 4z = 40  (Do you see how the \"X\" variables were eliminated?)\r<BR>\n" );
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document.write( "Now let's take another two equations and eliminate the SAME variable.  I'd suggest the 2nd and 3rd variable.  Let's eliminate for \"X\" because that is what we eliminated above......\r<BR>\n" );
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document.write( "5X + Y + 4Z = 15<BR>\n" );
document.write( "X +2y -3Z = 12 (Hmm.. what if we multiplied this whole equation by -5)?\r<BR>\n" );
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document.write( "If we multiply the 2nd equation by -5, the two equations will then be....\r<BR>\n" );
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document.write( "5X + Y + 4Z = 15<BR>\n" );
document.write( "-5x- 10y+15z = -60<BR>\n" );
document.write( "__________________   Now let's add these.........<BR>\n" );
document.write( "-9y + 19z = -45\r<BR>\n" );
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document.write( "OHhhh.. now we have two equations that have two variables.  Let's work with them...........\r<BR>\n" );
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document.write( "8y + 4z = 40  (Hm... what if we multiply this whole line by 9?) <BR>\n" );
document.write( "-9y + 19z = -45 (What if we multiply this whole line by 8?)\r<BR>\n" );
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document.write( "If we did the multiplications I suggest above, we'd get:\r<BR>\n" );
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document.write( "72y + 36z = 360<BR>\n" );
document.write( "-72y + 152z = -360<BR>\n" );
document.write( "_____________________  Let's add these equations....<BR>\n" );
document.write( "0y   + 188z = 0<BR>\n" );
document.write( "z = 0 (divided both sides by 188)\r<BR>\n" );
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document.write( "Now we know that z = 0.  Let's plug that number into one of our 2 variable equations......\r<BR>\n" );
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document.write( "8y + 4z = 40<BR>\n" );
document.write( "8y + 4(0) = 40<BR>\n" );
document.write( "8y = 40<BR>\n" );
document.write( "y = 5 (divided both sides by 8)\r<BR>\n" );
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document.write( "Now we know that z = 0 and y = 5.  Let's plug those numbers into one of our 3 variable equations..............\r<BR>\n" );
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document.write( "X + 2y -3Z = 12  (one of our original 3 variable equations)<BR>\n" );
document.write( "x + 2(5) -3(0)= 12 (plugged in our numbers for variables)<BR>\n" );
document.write( "x + 10 = 12 (multiplied 2 times 5 and -3 times 0)<BR>\n" );
document.write( "x = 12 - 10 (subtracted 10 from both sides)<BR>\n" );
document.write( "x = 2\r<BR>\n" );
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document.write( "We now know that x = 2, y = 5 and z = 0.   If we have worked correctly, we can plug these in for the variables on our original three equations.  Let's check.........\r<BR>\n" );
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document.write( "Original equation 1\r<BR>\n" );
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document.write( "X+10Y+Z=52 (original equation)<BR>\n" );
document.write( "2 + 10(5) + 0 = 52<BR>\n" );
document.write( "2 + 50 + 0 = 52<BR>\n" );
document.write( "52 = 52<BR>\n" );
document.write( "Yay! The 1st equation works out correctly.\r<BR>\n" );
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document.write( "Second original equation\r<BR>\n" );
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document.write( "5X+Y+4Z=15<BR>\n" );
document.write( "5(2) + 5 + 4(0)= 15<BR>\n" );
document.write( "10 + 5 + 0 = 15<BR>\n" );
document.write( "15 = 15<BR>\n" );
document.write( "Yay again!  This 2nd equation works out correctly.  Let's go for the 3rd!\r<BR>\n" );
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document.write( "Third original equation\r<BR>\n" );
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document.write( "X+2y-3Z=12<BR>\n" );
document.write( "2 + 2(5) - 3(0) = 12<BR>\n" );
document.write( "2 + 10 - 0 = 12<BR>\n" );
document.write( "12 = 12<BR>\n" );
document.write( "Yayyyyy for the third right answer.   So our x, y, and z values are correct. \r<BR>\n" );
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document.write( "I hope this helps. :-) </SPAN>\n" );
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