document.write( "Question 155120: We have been studying the system Linear Equations in Three Variables by elimination method of x,y,z and I'm lost. PLEASE help!
\n" ); document.write( "x+y+z=4
\n" ); document.write( "x-y+2z=1
\n" ); document.write( "x-y-3z=-4 \n" ); document.write( "

Algebra.Com's Answer #114213 by Edwin McCravy(20056) $\"\"$ $\"About$

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x+y+z=4
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\r\n" );
document.write( "  x + y +  z =  4\r\n" );
document.write( "  x - y + 2z =  1\r\n" );
document.write( "  x - y - 3z = -4\r\n" );
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document.write( "1. If there happens to be one or more equations which contains \r\n" );
document.write( "   only two of the 3 unknowns, then write it down and skip to \r\n" );
document.write( "   step 3.\r\n" );
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document.write( "2. Choose two of the equations and an unknown which they both \r\n" );
document.write( "   contain to eliminate. Then eliminate that unknown from them.\r\n" );
document.write( "   Then write the resulting equation down.\r\n" );
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document.write( "3. Pick a pair of equations which both contain the SAME letter\r\n" );
document.write( "   that is missing in the equation you have written down, either\r\n" );
document.write( "   from step 1 or 2.  Then use multiplication to eliminate that\r\n" );
document.write( "   SAME unknown from them. Write that resulting equation under\r\n" );
document.write( "   the other equation.\r\n" );
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document.write( "4. Now you have a system of two equations in two unknowns. Solve\r\n" );
document.write( "   that for both unknowns.\r\n" );
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document.write( "5. Find the third unknown by substituting the values found in step\r\n" );
document.write( "   4 into any one of the original equations which contains it.\r\n" );
document.write( "    \r\n" );
document.write( "Now we'll go through those rules:\r\n" );
document.write( "\r\n" );
document.write( "  x + y +  z =  4\r\n" );
document.write( "  x - y + 2z =  1\r\n" );
document.write( "  x - y - 3z = -4\r\n" );
document.write( "\r\n" );
document.write( "1. If there happens to be one equation which contains only \r\n" );
document.write( "   two of the unknowns, then write it down and skip to step 3.\r\n" );
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document.write( "There is no equation with fewer than 3 unknowns so we cannot\r\n" );
document.write( "skip step 2.\r\n" );
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document.write( "2. Choose two of the equations and an unknown which they both \r\n" );
document.write( "   contain to eliminate. \r\n" );
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document.write( "Let's choose the first and second equations and choose y to\r\n" );
document.write( "eliminate because that's the easiest to eliminate:\r\n" );
document.write( "\r\n" );
document.write( "  x + y +  z =  4\r\n" );
document.write( "  x - y + 2z =  1\r\n" );
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document.write( "We just add them as they are and eliminate the y's\r\n" );
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document.write( " 2x     + 3z =  5\r\n" );
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document.write( "Then we write the resulting equation down.\r\n" );
document.write( "\r\n" );
document.write( "     2x + 3z = 5\r\n" );
document.write( "\r\n" );
document.write( "3. Pick a pair of equations which both contain the SAME letter\r\n" );
document.write( "   that is missing in the equation you have written down, either\r\n" );
document.write( "   from step 1 or 2.  Then eliminate that SAME unknown from them. \r\n" );
document.write( "  \r\n" );
document.write( "Let's pick the first and third original equations, and we must eliminate\r\n" );
document.write( "the SAME variable, y, from them\r\n" );
document.write( "\r\n" );
document.write( "  x + y +  z =  4\r\n" );
document.write( "  x - y - 3z = -4\r\n" );
document.write( "\r\n" );
document.write( "We just add them as they are and eliminate the y's\r\n" );
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document.write( "\r\n" );
document.write( "\r\n" );
document.write( " 2x    -  2z =  0\r\n" );
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document.write( "Then write the resulting equation under\r\n" );
document.write( "   the other equation.\r\n" );
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document.write( "    2x + 3z = 5\r\n" );
document.write( "    2x - 2z = 0\r\n" );
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document.write( "4. Now you have a system of two equations in two unknowns.\r\n" );
document.write( "   Solve that system for both unknowns.\r\n" );
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document.write( "I assume you know how to solve that system either by \r\n" );
document.write( "elimination or substitution. \r\n" );
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document.write( "The solution to that is x=1 and z=1\r\n" );
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document.write( "5. Find the third unknown, y, by substituting the values \r\n" );
document.write( "   found in step 4 into any one of the original equations \r\n" );
document.write( "   which contains it.  Solve for that letter.\r\n" );
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document.write( "Let's choose the first original equation:\r\n" );
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document.write( "      x + y + z = 4\r\n" );
document.write( "  (1) + y + (1) = 4\r\n" );
document.write( "      1 + y + 1 = 4\r\n" );
document.write( "          2 + y = 4\r\n" );
document.write( "              y = 2\r\n" );
document.write( "                \r\n" );
document.write( "So the solution is (x, y, z) = (1, 2, 1)\r\n" );
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document.write( "Edwin

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