document.write( "Question 25003: Instructions were to \"Solve the system of linear equations\"
\n" ); document.write( "(There is a bracket( { ) on one side of all three equations)\r
\n" ); document.write( "\n" ); document.write( "-x + y + 2z = 1
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\n" ); document.write( "5x + 4y + 2z = 4\r
\n" ); document.write( "\n" ); document.write( "I have read the section explaining this and am still unsure how to start. The other systems that include only two equations were not hard. \n" ); document.write( "
Algebra.Com's Answer #13419 by AnlytcPhil(1806)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "-x +  y + 2z =  1\r\n" );
document.write( "2x + 3y +  z = -2\r\n" );
document.write( "5x + 4y + 2z =  4\r\n" );
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document.write( "If we have this system of equations, \r\n" );
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document.write( "Ax + By + Cz = D\r\n" );
document.write( "Ex + Fy + Gz = H\r\n" );
document.write( "Ix + Jy + Kz = L\r\n" );
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document.write( "We want to change it to a system that has this form:\r\n" );
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document.write( "Mx + Ny + Pz = Q\r\n" );
document.write( "     Ry + Sz = T\r\n" );
document.write( "          Uz = V \r\n" );
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document.write( "So we want to get \r\n" );
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document.write( "eq (1)  -x +  y + 2z =  1\r\n" );
document.write( "eq (2)  2x + 3y +  z = -2\r\n" );
document.write( "eq (3)  5x + 4y + 2z =  4\r\n" );
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document.write( "that way.  To eliminate the 2x, we multiply eq (1) thru by 2\r\n" );
document.write( "and add it to 1 times eq (2).  \r\n" );
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document.write( "2[-x +  y + 2z =  1]   --- >   -2x + 2y + 4z =  2\r\n" );
document.write( "1[2x + 3y +  z = -2]   --- >    2x + 3y +  z = -2\r\n" );
document.write( "  5x + 4y + 2z =  4            覧覧覧覧覧覧覧覧覧予r\n" );
document.write( "                                     5y + 5z =  0\r\n" );
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document.write( "We observe that we can have a simpler equation by dividing\r\n" );
document.write( "this equation through by 5:           y +  z =  0 \r\n" );
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document.write( "So we write that equation as the new eq (2). Now we have\r\n" );
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document.write( "eq (1)  -x +  y + 2z =  1\r\n" );
document.write( "eq (2)        y +  z =  0\r\n" );
document.write( "eq (3)  5x + 4y + 2z =  4\r\n" );
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document.write( "Next we eliminate the 5x. To do that, we multiply eq (1) \r\n" );
document.write( "thru by -5 and add it to 1 times eq (3).  \r\n" );
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document.write( "5[-x +  y + 2z = 1]    --- >   -5x + 5y + 10z = 5\r\n" );
document.write( "        y +  z = 0   \r\n" );
document.write( "1[5x + 4y + 2z = 4]    --- >    5x + 4y +  2z = 4         \r\n" );
document.write( "                                     9y + 12z = 9\r\n" );
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document.write( "We observe that we can have a simpler equation by dividing\r\n" );
document.write( "this equation through by 3:          3y +  4z = 3 \r\n" );
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document.write( "So we write that equation as the new eq (3). Now we have\r\n" );
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document.write( "eq (1)  -x +  y + 2z = 1\r\n" );
document.write( "eq (2)        y +  z = 0\r\n" );
document.write( "eq (3)       3y + 4z = 3\r\n" );
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document.write( "Finally we eliminate the 3y. To do that, we multiply eq (2) \r\n" );
document.write( "thru by -3 and add it to 1 times eq (3).  \r\n" );
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document.write( "   -x +  y + 2z = 1    \r\n" );
document.write( "-3[      y +  z = 0]    --- >        -3y - 3z = 0      \r\n" );
document.write( " 1[     3y + 4z = 3]    --- >         3y + 4z = 3         \r\n" );
document.write( "                                            z = 3\r\n" );
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document.write( "So we write that equation as the new eq (3). Now we have\r\n" );
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document.write( "eq (1)  -x +  y + 2z = 1\r\n" );
document.write( "eq (2)        y +  z = 0\r\n" );
document.write( "eq (3)             z = 3\r\n" );
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document.write( "Now all we need to do is do \"back substitution\".\r\n" );
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document.write( "Since eq (3) tells us that z = 3, we replace z by 3 in eq (2)\r\n" );
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document.write( "              y + z =  0\r\n" );
document.write( "              y + 3 =  0\r\n" );
document.write( "                  y = -3\r\n" );
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document.write( "Now that we know both y and z, we can substitute those in eq (1)\r\n" );
document.write( "and find x\r\n" );
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document.write( "              -x + y + 2z =  1\r\n" );
document.write( "         -x + (-3) + 2(3) =  1\r\n" );
document.write( "               -x - 3 + 6 =  1 \r\n" );
document.write( "                   -x + 3 =  1\r\n" );
document.write( "                       -x = -2\r\n" );
document.write( "                        x =  2\r\n" );
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document.write( "So the solution is (x, y, z) = (2, -3, 3)\r\n" );
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document.write( "Edwin\r\n" );
document.write( "AnlytcPhil@aol.com
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