document.write( "Question 153681: I do not understand how to solve systems of linear equations in three variables.\r
\n" ); document.write( "\n" ); document.write( " Could someone please explain how I would solve this problem\r
\n" ); document.write( "\n" ); document.write( " x+y-z=-1
\n" ); document.write( " -4x+-y+2z=-7
\n" ); document.write( " 2x+-2y+-5z=7 \n" ); document.write( "
Algebra.Com's Answer #113131 by ktsau(1)\"\" \"About 
You can put this solution on YOUR website!
Hi! We usually use simultaneous equations to solve this kind of question.\r
\n" ); document.write( "\n" ); document.write( "Firstly, name the formulas:\r
\n" ); document.write( "\n" ); document.write( "x+y-z=-1 -- (1)
\n" ); document.write( "-4x+-y+2z=-7 -- (2)
\n" ); document.write( "2x+-2y+-5z=7 -- (3)\r
\n" ); document.write( "\n" ); document.write( "the next thing we do is to prevent those three variables appear in a same formula, so, we need to remove one of them by subtracting or adding the formulas together. At the same time, we need to express a variable in terms of another variable. If you look careful enough, you will find 'y' can be removed by adding (1) and (2) together:\r
\n" ); document.write( "\n" ); document.write( "(1)+(2): x+y-z+(-4x-y+2z) = -1+(-7)
\n" ); document.write( " x-4x+y-y-z+2z = -8
\n" ); document.write( " -3x+z = -8
\n" ); document.write( " z = 3x-8 -- (4)\r
\n" ); document.write( "\n" ); document.write( "As shown above, i've expressed z in terms of x, and a new formula has produced. What i usually do is to give a name for new formulas: (4). Okay, let's ignore (4) for a second. And take a look at (2) and (3), i'm going to remove 'y' by substracting. But before that, we need to multiply (2) by 2, otherwise, the 'y' won't go.\r
\n" ); document.write( "\n" ); document.write( "(2)x2: (-4x+-y+2z)x2=-7x2
\n" ); document.write( " -8x-2y+4z =-14 -- (5)\r
\n" ); document.write( "\n" ); document.write( "Now, we can remove 'y' by substracting:\r
\n" ); document.write( "\n" ); document.write( "(5)-(3): -8x-2y+4z-(2x-2y-5z) = -14-(7)
\n" ); document.write( " -8x-2x-2y+2y+4z+5z = -21
\n" ); document.write( " -10x+9z = -21\r
\n" ); document.write( "\n" ); document.write( "At this moment, we substitute (4) into above formula.\r
\n" ); document.write( "\n" ); document.write( " -10x+9(3x-8) = -21
\n" ); document.write( " -10x+27x-72 = -21
\n" ); document.write( " 17x = -21+72
\n" ); document.write( " 17x = 51
\n" ); document.write( " x = 3\r
\n" ); document.write( "\n" ); document.write( "Now we've got one variable solved. Let's use 'x' to solve the others!
\n" ); document.write( "In here, we substitute x=3 into (4):
\n" ); document.write( " z=3(3)-8
\n" ); document.write( " z=1\r
\n" ); document.write( "\n" ); document.write( "Here's the last step, substitute all the knowns into one formula, i'll use the easiest one.\r
\n" ); document.write( "\n" ); document.write( "Sub. x=3, z=1 into (1)\r
\n" ); document.write( "\n" ); document.write( " (3)+y-(1) =-1
\n" ); document.write( " y =-1-3+1
\n" ); document.write( " y =-3\r
\n" ); document.write( "\n" ); document.write( "so, x=3, y=-3 and z=1. You may also check the answers by substituting them into other formulas. ( From (2): -4(3)-(-3)+2(1) = -7 )\r
\n" ); document.write( "\n" ); document.write( "Hope it helps!\r
\n" ); document.write( "\n" ); document.write( "see you later!
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