SOLUTION: Solve by substitution or elimination: 2x-3y+z=14 4x+2y-5z=-3 x+y+2z=1

Question 162864: Solve by substitution or elimination:
2x-3y+z=14
4x+2y-5z=-3
x+y+2z=1
Answer by elima(1433) (Show Source): You can put this solution on YOUR website!
I'll use elimination to solve;
2x-3y+z=14
4x+2y-5z=-3
x+y+2z=1
==================
First add two equations together to eliminate a variable;
4x+2y-5z=-3 - multiply equation by 2 so you can eliminate the z;
x+y+2z=1 - multiply equation by 5 so you can eliminate the z;
=========================
2(4x+2y-5s=-3)
8x+4y-10z=-6
5(x+y+2z=1)
5x+5y+10z=5
===============
8x+4y-10z=-6
5x+5y+10z=5
-----------
13x+9y=-1
====================
Now choose two others to eliminate the z;
2x-3y+z=14 - multiply by -2
x+y+2z=1
=======================
-2(2x-3y+z=14)
-4x+6y-2z=-28
x+y+2z=1
------------
-3x+7y=-27
====================
Now add the two new equations together to eliminate another variable;
13x+9y=-1 - multiply by -7
-3x+y=-27 - multiply by 9
============================
-7(13x+9y=-1)
-91x-63y=-14
9(-3x+7y=-27)
-21x+63y=-243
=============================
-91x-63y=7
-27x+63y=-243
-------------
-118x=-236
x=2
=======================
Now that we have the x, let's plug it into one of the new equations and solve for y;
13(2)+9y=-1
26+9y=-1
9y=-27
y=-3
===========================
We have our x and y, let's plug them in and get the z;
x+y+2z=1
2+-3+2z=1
-1+2z=1
2z=2
z=1
=========================
check;
2x-3y+z=14
2(2)-3(-3)+1=14
4+9+1=14
14=14
check;
4x+2y-5z=-3
4(2)+2(-3)-5(1)=-3
8-6-5=-3
-3=-3
:)

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