document.write( "Question 101338: Could you please take the time to help me solve this question: Solve the following system of three linear equations: 2x + y + z = 3, -x + 2y + 2z = 1 and x - y - 3z = -6. Noting that each equation represents a plane, interpret your results.\r
\n" ); document.write( "\n" ); document.write( "Thank you, very much for taking the time to help me, most appreciated \n" ); document.write( "

Algebra.Com's Answer #73818 by MathLover1(20850) $\"\"$ $\"About$

You can put this solution on YOUR website!
first take these two equations:
\n" ); document.write( " $\"2x+%2B+y+%2B+z+=+3\"$
\n" ); document.write( " $\"-x+%2B+2y+%2B+2z+=+1\"$ (multiply both sides by $\"2\"$ in this one)
\n" ); document.write( "then we will have
\n" ); document.write( " $\"2x+%2B+y+%2B+z+=+3\"$
\n" ); document.write( " $\"-2x+%2B+4y+%2B+4z+=+2\"$
\n" ); document.write( "now we will eliminate $\"x\"$ by adding them and it will be:\r
\n" ); document.write( "\n" ); document.write( " $\"%28y%2B4y%29+%2B%28z%2B4z%29=3%2B2\"$
\n" ); document.write( " $\"5y+%2B5z+=5\"$ now divide both sides by $\"5\"$ ,and we will have:\r
\n" ); document.write( "\n" ); document.write( " $\"y%2Bz=1\"$
\n" ); document.write( "then we have
\n" ); document.write( " $\"y=-z%2B1\"$ which we can put in third equation
\n" ); document.write( " $\"x+-+y+-+3z+=+-6\"$
\n" ); document.write( " and we will have
\n" ); document.write( "
\n" ); document.write( " $\"x+%2Bz+-1+-+3z+=+-6\"$
\n" ); document.write( " $\"x+%2Bz+-1+-3z+=-6\"$
\n" ); document.write( " $\"x+-2z+=-6+%2B1\"$
\n" ); document.write( " $\"x+-2z+=-5\"$ \r
\n" ); document.write( "\n" ); document.write( "from this equation
\n" ); document.write( " $\"x=2z-5\"$ \r
\n" ); document.write( "\n" ); document.write( "since we have $\"x+=+2z+-+5\"$ and $\"y+=+-z+%2B+1\"$ , we can use them to calculate $\"z\"$ :\r
\n" ); document.write( "\n" ); document.write( "let use the second equation
\n" ); document.write( " $\"-x+%2B+2y+%2B+2z+=+1\"$
\n" ); document.write( " substitute $\"x\"$ and $\"y+\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"-%282z-5%29+%2B+2%28-z+%2B+1%29%2B+2z+=+1\"$
\n" ); document.write( " $\"-2z+%2B+5+-2z+%2B+2+%2B+2z+=+1\"$
\n" ); document.write( " $\"-2z=+-6\"$
\n" ); document.write( " $\"z=-6%2F-2\"$
\n" ); document.write( " $\"z=3+\"$
\n" ); document.write( "now is easy to calculate $\"x\"$ and $\"y\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x=+2z+%96+5\"$
\n" ); document.write( " $\"x+=+2%2A3+%96+5\"$
\n" ); document.write( " $\"x+=+1\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"y+=+-z+%2B+1\"$
\n" ); document.write( " $\"y+=+-3+%2B+1\"$
\n" ); document.write( " $\"y+=+-2\"$
\n" ); document.write( " \n" ); document.write( "

\n" );