document.write( "Question 1196171: How do you solve this system of three linear equations using elimination method
\n" ); document.write( "3x+2y+z=1
\n" ); document.write( "x+y+z=0
\n" ); document.write( "5x+3y-2z=-4 \n" ); document.write( "
Algebra.Com's Answer #828888 by math_tutor2020(3816)\"\" \"About 
You can put this solution on YOUR website!

\n" ); document.write( "I'll get you started.\r
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\n" ); document.write( "\n" ); document.write( "There are a multiple number of ways to approach elimination problems.
\n" ); document.write( "It's mostly a trial and error type of thing, or something you develop a good eye for once you get enough practice.\r
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\n" ); document.write( "\n" ); document.write( "Let's eliminate the variable z
\n" ); document.write( "The first two equations have +z in them
\n" ); document.write( "If we flip the signs of each term in equation (2), then we go from x+y+z = 0 to -x-y-z = 0\r
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\n" ); document.write( "\n" ); document.write( "So we have
\n" ); document.write( "3x+2y+z = 1
\n" ); document.write( "-x-y-z = 0\r
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\n" ); document.write( "\n" ); document.write( "Add the equations straight down
  • The x terms add to 3x + (-x) = 2x
  • The y terms add to 2y + (-y) = y
  • The z terms cancel because z + (-z) = 0z = 0, which we intended (and why we did the sign flip in the second equation)
  • The right hand sides add to 1+0 = 1
We end up with 2x+y = 1
\n" ); document.write( "I'll call this equation (4)\r
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\n" ); document.write( "\n" ); document.write( "Return back to the original system
\n" ); document.write( "3x+2y+z=1
\n" ); document.write( "x+y+z=0
\n" ); document.write( "5x+3y-2z=-4\r
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\n" ); document.write( "\n" ); document.write( "Temporarily delete the first equation to get
\n" ); document.write( "x+y+z=0
\n" ); document.write( "5x+3y-2z=-4\r
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\n" ); document.write( "\n" ); document.write( "Now if we were to double everything in equation (2), then we go from x+y+z = 0 to 2x+2y+2z = 0\r
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\n" ); document.write( "\n" ); document.write( "So this system
\n" ); document.write( "x+y+z=0
\n" ); document.write( "5x+3y-2z=-4\r
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\n" ); document.write( "\n" ); document.write( "is the same as
\n" ); document.write( "2x+2y+2z=0
\n" ); document.write( "5x+3y-2z=-4\r
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\n" ); document.write( "\n" ); document.write( "Add straight down:
  • The x terms add to 2x + 5x = 7x
  • The y terms add to 2y + 3y = 5y
  • The z terms cancel because 2z + (-2z) = 0z = 0
  • The right hand sides add to 0 + (-4) = -4
We end up with:
\n" ); document.write( "7x+5y = -4
\n" ); document.write( "which I'll refer to as equation (5)\r
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\n" ); document.write( "\n" ); document.write( "We now have a smaller system of equations
\n" ); document.write( "2x+y = 1
\n" ); document.write( "7x+5y = -4
\n" ); document.write( "which were equations (4) and (5) mentioned earlier.\r
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\n" ); document.write( "\n" ); document.write( "I'll let you finish up the problem. Feel free to ask about any step, or if you are still stuck. \r
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\n" ); document.write( "\n" ); document.write( "Hint: multiply equation (4) by -5 so you can eliminate the y variable.
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