document.write( "Question 62306: I am having trouble solving these systems. I get lost somewhere in the middle and it becomes a mess. Please help.
\n" ); document.write( "x-2y-3z=4
\n" ); document.write( "2x-4y+5z=-3
\n" ); document.write( "5x-6y+4z=-7\r
\n" ); document.write( "\n" ); document.write( "and\r
\n" ); document.write( "\n" ); document.write( "2x-y=-1
\n" ); document.write( "-2x+z=1
\n" ); document.write( "y-z=0\r
\n" ); document.write( "\n" ); document.write( "Thank you. \n" ); document.write( "
Algebra.Com's Answer #43200 by ankor@dixie-net.com(22740)\"\" \"About 
You can put this solution on YOUR website!
You can solve this as a matrix, but you can also elimination and substitution:
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\n" ); document.write( "x - 2y - 3z = 4
\n" ); document.write( "2x -4y + 5z = -3
\n" ); document.write( "5x -6y + 4z = -7
\n" ); document.write( ":
\n" ); document.write( "Let's just deal with the 1st & 2nd equation:
\n" ); document.write( "x - 2y - 3z = 4
\n" ); document.write( "2x -4y + 5z = -3
\n" ); document.write( ":
\n" ); document.write( "Mult the 1st equation by 2, leave the 2nd equation as it is:
\n" ); document.write( "2x - 4y - 6z = 8
\n" ); document.write( "2x - 4y + 5z = -3
\n" ); document.write( "---------------------subtract & we eliminate x & y, it's easy to find z!
\n" ); document.write( "0x + 0y -11z = 11
\n" ); document.write( "z = 11/-11
\n" ); document.write( "z = -1
\n" ); document.write( ":
\n" ); document.write( "Now that we know z = -1 we can rewrite the 3 equations:
\n" ); document.write( "x - 2y - 3(-1) = 4 >> x -2y + 3 = 4 >> x - 2y = 4 - 3
\n" ); document.write( "2x -4y + 5(-1) = -3 >> 2x -4y - 5 = -3 >> 2x -4y = -3 + 5
\n" ); document.write( "5x -6y + 4(-1) = -7 >> 5x -6y - 4 = -7 >> 5x - 6y = -7 + 4
\n" ); document.write( "We end up with 3 equation with 2 unknowns:
\n" ); document.write( "x - 2y = +1
\n" ); document.write( "2x -4y = +2
\n" ); document.write( "5x -6y = -3
\n" ); document.write( ":
\n" ); document.write( "Use the 1st equation for substitution:
\n" ); document.write( "x = 2y + 1
\n" ); document.write( ":
\n" ); document.write( "Substitute for x in the 3rd \"2 unknown\" equation:
\n" ); document.write( "5(2y+1) - 6y = -3
\n" ); document.write( "10y + 5 - 6y = -3
\n" ); document.write( "4y = -3 - 5
\n" ); document.write( "4y = -8
\n" ); document.write( "y = -8/4
\n" ); document.write( "y = -2
\n" ); document.write( ":
\n" ); document.write( "Find x using x = 2y +1
\n" ); document.write( "x = 2(-2) + 1
\n" ); document.write( "x = -4 + 1
\n" ); document.write( "x = -3
\n" ); document.write( ":
\n" ); document.write( "We have x = -3, y = -2; z = -1
\n" ); document.write( "Choose one of the original equations to check our solutions, the 2nd equation:
\n" ); document.write( "2x -4y + 5z = -3
\n" ); document.write( "2(-3) - 4(-2) + 5(-1) = -3
\n" ); document.write( "-6 + 8 - 5 = -3; equality reigns
\n" ); document.write( ":
\n" ); document.write( ":
\n" ); document.write( ":
\n" ); document.write( "and
\n" ); document.write( "2x - y = -1
\n" ); document.write( "-2x + z = 1
\n" ); document.write( "y - z = 0
\n" ); document.write( ":
\n" ); document.write( "Look at the equation y - z = 0, that means y = z, right,
\n" ); document.write( "Let's replace y with z in the 1st equation and pair it with the 2nd equation:
\n" ); document.write( "2x - z = -1
\n" ); document.write( "-2x + z = 1
\n" ); document.write( "------------ add these and you get
\n" ); document.write( "0 + 0 = 0
\n" ); document.write( ":
\n" ); document.write( "That means there is no unique solution to this system
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