document.write( "Question 1090977: Can someone help me?
\n" ); document.write( "I need to write the final answer as an ordered triple (x,y,z)
\n" ); document.write( "Solve the system for x,y,z
\n" ); document.write( "2x+3y-z=4
\n" ); document.write( "3x-y+2z=5
\n" ); document.write( "x-4y+3z=1
\n" ); document.write( "I tried it, but ended up weird.
\n" ); document.write( "Thank you. \n" ); document.write( "

Algebra.Com's Answer #705432 by MathLover1(20850) $\"\"$ $\"About$

You can put this solution on YOUR website!
Solve the system for $\"x\"$ , $\"y\"$ , $\"z\"$
\n" ); document.write( " $\"2x%2B3y-z=4\"$ .....eq.1
\n" ); document.write( " $\"3x-y%2B2z=5\"$ .....eq.2
\n" ); document.write( " $\"x-4y%2B3z=1\"$ .....eq.3\r
\n" ); document.write( "\n" ); document.write( "------------------------------------------
\n" ); document.write( " $\"2x%2B3y-z=4\"$ .....eq.1...solve for $\"z\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"z=2x%2B3y-4\"$ .....eq.(1z)\r
\n" ); document.write( "\n" ); document.write( " $\"3x-y%2B2z=5\"$ .....eq.2..solve for $\"z\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"2z=-3x%2By%2B5\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"z=-3x%2F2%2By%2F2%2B5%2F2\"$ ....eq.(2z)\r
\n" ); document.write( "\n" ); document.write( " $\"x-4y%2B3z=1\"$ .....eq.3..........solve for $\"z\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"3z=-x%2B4y1\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"z+=+-x%2F3+%2B+%284+y%29%2F3+%2B+1%2F3\"$ ....eq.(3z)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "from eq.(1z) and eq.(2z):\r
\n" ); document.write( "\n" ); document.write( " $\"2x%2B3y-4=-3x%2F2%2By%2F2%2B5%2F2\"$ ...solve for $\"x\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"4x%2B6y-8=-3x%2By%2B5\"$
\n" ); document.write( " $\"4x%2B3x=y-6y%2B5%2B8\"$
\n" ); document.write( " $\"7x=-5y%2B5%2B8\"$
\n" ); document.write( " $\"x=-5y%2F7%2B13%2F7\"$ .....(a)\r
\n" ); document.write( "\n" ); document.write( "from eq.(2z) and eq.(3z):\r
\n" ); document.write( "\n" ); document.write( " $\"-3x%2F2%2By%2F2%2B5%2F2=-x%2F3+%2B+%284+y%29%2F3+%2B+1%2F3\"$ ....both sides multiply by $\"6\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"-9x%2B3y%2B15=-2x+%2B+8+y+%2B+2\"$
\n" ); document.write( " $\"-8y%2B3y%2B15-2=-2x+%2B+9x\"$
\n" ); document.write( " $\"-5y%2B13=7x\"$
\n" ); document.write( " $\"x=-5y%2F7%2B13%2F7\"$ .....(b)\r
\n" ); document.write( "\n" ); document.write( " $\"3x-y%2B2z=5\"$ .....eq.2....substitute $\"x\"$ and $\"z\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"3%28-5y%2F7%2B13%2F7%29-y%2B2%28-3x%2F2%2By%2F2%2B5%2F2%29=5\"$ .....eq.2\r
\n" ); document.write( "\n" ); document.write( " $\"highlight%28y+=+13%2F5+-+%287+x%29%2F5%29\"$ -> real solution\r
\n" ); document.write( "\n" ); document.write( " $\"z=2x%2B3y-4\"$ .....eq.(1z)......substitute $\"y\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"z=2x%2B3%2813%2F5+-+%287x%29%2F5%29-4\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"highlight%28z++=+19%2F5+-+%2811x%29%2F5%29\"$ -> real solution\r
\n" ); document.write( "\n" ); document.write( " $\"highlight%28x+=+5n+%2B+4%29\"$ , $\"highlight%28y+=+-7n+-+3%29\"$ , $\"highlight%28z+=+-11n+-+5%29\"$ , where $\"n\"$ element $\"Z\"$ \r
\n" ); document.write( "\n" ); document.write( "integer solutions:\r
\n" ); document.write( "\n" ); document.write( "( $\"x\"$ , $\"y\"$ , $\"z\"$ )=( $\"5n+%2B+4\"$ , $\"-7n+-+3\"$ , $\"-11n+-+5\"$ )\r
\n" ); document.write( "\n" ); document.write( "find some real take real solutions and equal to zero:\r
\n" ); document.write( "\n" ); document.write( " $\"+y=13%2F5+-+%287+x%29%2F5\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"+13%2F5+-+%287+x%29%2F5=0\"$ ...solve for $\"x\"$
\n" ); document.write( " $\"+13+-+7x=0\"$
\n" ); document.write( " $\"+13+=7x\"$
\n" ); document.write( " $\"highlight%28x=13%2F7%29\"$ \r
\n" ); document.write( "\n" ); document.write( "plug in\r
\n" ); document.write( "\n" ); document.write( " $\"z=19%2F5+-+%2811x%29%2F5%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"z=19%2F5+-+%2811%2813%2F7%29%29%2F5%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"highlight%28z=-2%2F7%29\"$ \r
\n" ); document.write( "\n" ); document.write( "find $\"y\"$ :\r
\n" ); document.write( "\n" ); document.write( " $\"y+=+13%2F5+-+%287%2813%2F7%29%29%2F5\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"highlight%28y+=+0%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "real solutions:( $\"x\"$ , $\"y\"$ , $\"z\"$ )=( $\"13%2F7\"$ , $\"0\"$ , $\"-2%2F7\"$ )\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "or
\n" ); document.write( "if $\"n=0\"$
\n" ); document.write( "( $\"x\"$ , $\"y\"$ , $\"z\"$ )=( $\"+4\"$ , $\"+-+3\"$ , $\"+-+5\"$ )
\n" ); document.write( "if $\"n=1\"$
\n" ); document.write( "( $\"x\"$ , $\"y\"$ , $\"z\"$ )=( $\"9\"$ , $\"-10\"$ , $\"-16\"$ ) ...and so on\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );