document.write( "Question 445237: I believe this topic explains it best, i need to solve the system using the elimination method. I have tried several times, and I feel like there is no solution; although my instructor said there is.\r
\n" ); document.write( "\n" ); document.write( "2w+x+y+2z=-2
\n" ); document.write( "3w+2x+y-z=9
\n" ); document.write( "w+x+y+z=0
\n" ); document.write( "w+2x+2y-z=10\r
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\n" ); document.write( "\n" ); document.write( "Thanks! \n" ); document.write( "

Algebra.Com's Answer #306772 by AnlytcPhil(1806) $\"\"$ $\"About$

You can put this solution on YOUR website!
I believe this topic explains it best, i need to solve the system using the elimination method. I have tried several times, and I feel like there is no solution; although my instructor said there is.
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document.write( "That's 4 equations in 4 variables\r\n" );
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document.write( "2w +  x +  y + 2z = -2\r\n" );
document.write( "3w + 2x +  y -  z =  9\r\n" );
document.write( " w +  x +  y +  z =  0\r\n" );
document.write( " w + 2x + 2y -  z = 10\r\n" );
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document.write( "First we get it down to 3 equations in 3 variables\r\n" );
document.write( "by choosing a variable to eliminate, and eliminating\r\n" );
document.write( "it from 3 pairs of equations making sure all 4 equations\r\n" );
document.write( "are used.  I will choose z to eliminate.\r\n" );
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document.write( "We eliminate z from the first two equations by\r\n" );
document.write( "multiplying the 2nd eq. by 2 and adding to the 1st:\r\n" );
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document.write( "2w +  x +  y + 2z = -2\r\n" );
document.write( "6w + 4x + 2y - 2z = 18\r\n" );
document.write( "----------------------\r\n" );
document.write( "8w + 5x + 3y      = 16\r\n" );
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document.write( "We eliminate z from the 2nd & 3rd equations by\r\n" );
document.write( "just adding them as they are:\r\n" );
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document.write( "3w + 2x +  y -  z =  9\r\n" );
document.write( " w +  x +  y +  z =  0\r\n" );
document.write( "----------------------\r\n" );
document.write( "4w + 3x + 2y      =  9\r\n" );
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document.write( "We eliminate z from the 3rd & 4th equations by\r\n" );
document.write( "just adding them as they are:\r\n" );
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document.write( "\r\n" );
document.write( " w +  x +  y +  z =  0\r\n" );
document.write( " w + 2x + 2y -  z = 10\r\n" );
document.write( "----------------------\r\n" );
document.write( "2w + 3x + 3y      = 10\r\n" );
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document.write( "Now we have it down to 3 equations in 3 variables.\r\n" );
document.write( "\r\n" );
document.write( "8w + 5x + 3y = 16\r\n" );
document.write( "4w + 3x + 2y =  9\r\n" );
document.write( "2w + 3x + 3y = 10\r\n" );
document.write( "\r\n" );
document.write( "Next we get it down to 2 equations in 2 variables\r\n" );
document.write( "by choosing a variable to eliminate, and eliminating\r\n" );
document.write( "it from 2 pairs of equations, making sure that all\r\n" );
document.write( "3 equations are used.  I choose w to eliminate:\r\n" );
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document.write( "We eliminate w from the 2nd & 3rd equations by multiplying\r\n" );
document.write( "the second equation by -2 and adding it to the 1st equation:\r\n" );
document.write( "\r\n" );
document.write( " 8w + 5x + 3y =  16\r\n" );
document.write( "-8w - 6x - 4y = -18\r\n" );
document.write( "-------------------\r\n" );
document.write( "      -x -  y =  -2\r\n" );
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document.write( "We eliminate w from the first two equations by multiplying\r\n" );
document.write( "the 3rd equation by -2 and adding it to the 2nd equation:\r\n" );
document.write( "\r\n" );
document.write( " 4w + 3x + 2y =   9\r\n" );
document.write( "-4w - 6x - 6y = -20\r\n" );
document.write( "-------------------\r\n" );
document.write( "     -3x - 4y = -11\r\n" );
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document.write( "Now we have it down to 2 equations in 2 variables.\r\n" );
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document.write( "      -x -  y =  -2\r\n" );
document.write( "     -3x - 4y = -11\r\n" );
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document.write( "Next we get it down to 1 equation in 1 variables\r\n" );
document.write( "by choosing a variable to eliminate.  I choose y \r\n" );
document.write( "to eliminate:\r\n" );
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document.write( "We eliminate y by multiplying the 1st equation by -4\r\n" );
document.write( "and adding the 2nd equation to it:\r\n" );
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document.write( "      4x + 4y =   8\r\n" );
document.write( "     -3x - 4y = -11\r\n" );
document.write( "   ----------------\r\n" );
document.write( "       x      =  -3\r\n" );
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document.write( "That's 1 equation in 1 variable solved. So we \r\n" );
document.write( "substitute in \r\n" );
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document.write( "      -x - y = -2\r\n" );
document.write( "   -(-3) - y = -2\r\n" );
document.write( "       3 - y = -2\r\n" );
document.write( "          -y = -5\r\n" );
document.write( "           y =  5 \r\n" );
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document.write( "Then we substitute x = -3 and y = 5 in\r\n" );
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document.write( "     2w + 3x + 3y = 10\r\n" );
document.write( "2w + 3(-3) + 3(5) = 10\r\n" );
document.write( "      2w - 9 + 15 = 10\r\n" );
document.write( "           2w + 6 = 10\r\n" );
document.write( "               2w = 4\r\n" );
document.write( "                w = 2 \r\n" );
document.write( "Then we substitute x = -3 and y = 5 and w = 2 in\r\n" );
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document.write( "    w + x + y + z =  0\r\n" );
document.write( " 2 + (-3) + 5 + z = 0\r\n" );
document.write( "            4 + z = 0\r\n" );
document.write( "                z = -4\r\n" );
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document.write( "So (w,x,y,z) = (2,-3,5,-4)\r\n" );
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document.write( "Edwin

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