document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198509>Question 1198509</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the system of equations.<BR>\n" );
document.write( "X-2y+z=-3<BR>\n" );
document.write( "-x+4y=10<BR>\n" );
document.write( "2x-y+6z=7\r<BR>\n" );
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document.write( "The correct answer is one of the following.  Which one?\r<BR>\n" );
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document.write( "A) no solution <BR>\n" );
document.write( "B) x=2, y=3, z=1<BR>\n" );
document.write( "C) x=-1, y=4,z=2<BR>\n" );
document.write( "D) x=3, y=2, z=-1<BR>\n" );
document.write( "E) x=-2, y=3, z=-4<BR>\n" );
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document.write( "Let's solve the 2nd equation for x.\r<BR>\n" );
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document.write( "-x+4y = 10<BR>\n" );
document.write( "-x = 10-4y<BR>\n" );
document.write( "x = -10+4y<BR>\n" );
document.write( "x = 4y-10\r<BR>\n" );
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document.write( "Then plug it into the 1st equation<BR>\n" );
document.write( "x-2y+z = -3<BR>\n" );
document.write( "4y-10-2y+z = -3<BR>\n" );
document.write( "2y+z = -3+10<BR>\n" );
document.write( "2y+z = 7\r<BR>\n" );
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document.write( "And do the same for the 3rd equation<BR>\n" );
document.write( "2x-y+6z = 7<BR>\n" );
document.write( "2(4y-10)-y+6z = 7<BR>\n" );
document.write( "8y-20-y+6z = 7<BR>\n" );
document.write( "7y+6z = 7+20<BR>\n" );
document.write( "7y+6z = 27\r<BR>\n" );
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document.write( "We now have this system of equations<BR>\n" );
document.write( "2y+z = 7<BR>\n" );
document.write( "7y+6z = 27\r<BR>\n" );
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document.write( "I'll now isolate z in the 1st equation of this new system.<BR>\n" );
document.write( "I'm picking on z because the coefficient here is 1.\r<BR>\n" );
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document.write( "2y+z = 7<BR>\n" );
document.write( "z = 7-2y\r<BR>\n" );
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document.write( "Then substitute it into the other equation and solve for y.<BR>\n" );
document.write( "7y+6z = 27<BR>\n" );
document.write( "7y+6(7-2y) = 27<BR>\n" );
document.write( "7y+42-12y = 27<BR>\n" );
document.write( "-5y+42 = 27<BR>\n" );
document.write( "-5y = 27-42<BR>\n" );
document.write( "-5y = -15<BR>\n" );
document.write( "y = -15/(-5)<BR>\n" );
document.write( "y = 3<BR>\n" );
document.write( "This narrows our choices between choice (B) or choice (E).\r<BR>\n" );
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document.write( "Use that value of y to find z<BR>\n" );
document.write( "z = 7-2y<BR>\n" );
document.write( "z = 7-2*3<BR>\n" );
document.write( "z = 7-6<BR>\n" );
document.write( "z = 1<BR>\n" );
document.write( "We can immediately conclude the answer is <font color=red>choice (B)</font>\r<BR>\n" );
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document.write( "If you wanted, let's keep going to find x.<BR>\n" );
document.write( "x = 4y-10<BR>\n" );
document.write( "x = 4*3-10<BR>\n" );
document.write( "x = 12-10<BR>\n" );
document.write( "x = 2<BR>\n" );
document.write( "This offers further confirmation that (B) is the answer.\r<BR>\n" );
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document.write( "If you are in a time crunch, then you can use the multiple choice answers to your advantage. <BR>\n" );
document.write( "Simply plug each numeric value in for the corresponding variable. Then simplify each side.\r<BR>\n" );
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document.write( "Let's say we wanted to check if (E) was the answer.<BR>\n" );
document.write( "We'd plug in x = -2, y = 3, z = -4 into the 1st equation to get...<BR>\n" );
document.write( "x-2y+z = -3<BR>\n" );
document.write( "-2-2(3)+(-4) = -3<BR>\n" );
document.write( "-2-6-4 = -3<BR>\n" );
document.write( "-12 = -3<BR>\n" );
document.write( "We get a false statement at the end. We need to get the same thing on both sides for it to be a true equation.<BR>\n" );
document.write( "Since the last equation is false, it causes the first equation to be false for those x,y,z values.<BR>\n" );
document.write( "Therefore, we can rule out choice (E) fairly quickly.<BR>\n" );
document.write( "Choices (C) and (D) are eliminated for similar reasoning.\r<BR>\n" );
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document.write( "To confirm choice (B) is the solution, plug (x,y,z) = (2,3,1) into each of the original three equations. <BR>\n" );
document.write( "You should get true results after simplification. <BR>\n" );
document.write( "I'll let you do these confirmations.\r<BR>\n" );
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document.write( "Answer: <font color=red>Choice (B)</font>  x=2, y=3, z=1<BR>\n" );
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