document.write( "Question 150608: pls.help me solve this equations involving 3 variables,using elimination method.
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Algebra.Com's Answer #110549 by Edwin McCravy(20060) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "3x -  y +  z =  2\r\n" );
document.write( "5x + 2y - 3z = 15\r\n" );
document.write( "2x +  y + 2z = 17\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" );
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document.write( "Now we'll go through those rules:\r\n" );
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document.write( "3x -  y +  z =  2\r\n" );
document.write( "5x + 2y - 3z = 15\r\n" );
document.write( "2x +  y + 2z = 17\r\n" );
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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 third equations and choose y to\r\n" );
document.write( "eliminate because that's the easiest to eliminate:\r\n" );
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document.write( "3x - y +  z =  2\r\n" );
document.write( "2x + y + 2z = 17\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( "5x     + 3z = 19\r\n" );
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document.write( "Then we write the resulting equation down.\r\n" );
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document.write( "    5x + 3z = 19\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 eliminate that SAME unknown from them. \r\n" );
document.write( "  \r\n" );
document.write( "Let's pick the first two original equations, and we must eliminate\r\n" );
document.write( "the SAME variable, y, from them\r\n" );
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document.write( "3x -  y +  z =  2\r\n" );
document.write( "5x + 2y - 3z = 15 \r\n" );
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document.write( "We multiply the upper equation by 2 so the y's will cancel out:\r\n" );
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document.write( " 6x - 2y + 2z =  4\r\n" );
document.write( " 5x + 2y - 3z = 15\r\n" );
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document.write( "We add them and eliminate the y's\r\n" );
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document.write( "11x     -   z = 19\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( "    5x + 3z = 19\r\n" );
document.write( "   11x -  z = 19\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=2 and z=3\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( "     3x - y + z =  2\r\n" );
document.write( " 3(2) - y + (3) =  2\r\n" );
document.write( "      6 - y + 3 =  2\r\n" );
document.write( "          9 - y =  2\r\n" );
document.write( "             -y = -7\r\n" );
document.write( "              y =  7    \r\n" );
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document.write( "So the solution is (x, y, z) = (2, 7, 3)\r\n" );
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document.write( "Edwin
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