SOLUTION: Use the elimination method to solve the following system of equations. x + 3y – z = 2 x – 2y + 3z = 7 x + 2y – 5z = –21

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Question 370091: Use the elimination method to solve the following system of equations. x + 3y – z = 2 x – 2y + 3z = 7
x + 2y – 5z = –21

Found 2 solutions by Fombitz, CharlesG2:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
1.x+%2B+3y+-z+=+2++
2.x+-2y+%2B+3z+=+7
3.x+%2B+2y+-+5z+=+-21
Subtract eq. 1 from eq. 2,
x-2y%2B3z-x-3y%2Bz=7-2
4.-5y%2B4z=5
Subtract eq. 1 from eq. 3,
x%2B2y-5z-x-3y%2Bz=-21-2
5.-y-4z=-23
Add eq. 4 and eq. 5,
-5y%2B4z-y-4z=5-23
-6y=-18
highlight%28+y=3%29
Then from eq. 5
-3-4z=-23
-4z=-20
highlight%28z=5%29
Then from eq. 1,
x+%2B+3%283%29+-+5+=+2++
x%2B4=2
highlight%28x=-2%29

Answer by CharlesG2(834) About Me  (Show Source):
You can put this solution on YOUR website!
Use the elimination method to solve the following system of equations.
x + 3y – z = 2
x – 2y + 3z = 7
x + 2y – 5z = –21

x + 3y - z = 2
x - 2y + 3z = 7
x + 2y - 5z = -21
pick a variable to eliminate, choosing x since all the x coefficients are the same, multiply the 1st equation by -1 and add it to the 2nd
-x - 3y + z = -2
0x - 5y + 4z = 5
x + 2y - 5z = -21
add 1st equation to 3rd equation
-x - 3y + z = -2
0x - 5y + 4z = 5
0x - y - 4z = -23
now all the coefficients on the x's are zeroed out save one,
multiply the 3rd equation by -5 and add the 2nd equation to it
-x - 3y + z = -2 --> -x - 3y + z = -2
0x - 5y + 4z = 5 --> 0x - 5y + 4z = 5
0x + 5y + 20z = 115 --> 0x + 0y + 24z = 120
multiply the 1st equation by 5, and the 2nd equation by -3, and add the 2nd equation to the first
-5x - 15y + 5z = -10 --> -5x + 0y - 7z = -25
0x + 15y - 12z = -15 --> 0x + 15y - 12z = -15
0x + 0y + 24z = 120 --> 0x + 0y + 24z = 120
multiply the 2nd equation by 2, add the 3rd equation to the 2nd
-5x + 0y - 7z = -25 --> -5x + 0y - 7z = -25
0x + 30y - 24z = -30 --> 0x + 30y + 0z = 90
0x + 0y + 24z = 120 --> 0x + 0y + 24z = 120
now we have 2 of the x coefficients zeroed out and 2 of the y coefficients zeroed out, now for the z's, multiply the first equation by 8, and the 3rd equation by 7/3, add the 3rd equation to the first
-40x + 0y - 56z = -200 --> -40x + 0y + 0z = 80
0x + 30y + 0z = 90 --> 0x + 30y + 0z = 90
0x + 0y + 56z = 280 --> 0x + 0y + 56z = 280
now divide the 1st equation by -40, the 2nd equation by 30,
and the 3rd equation by 56
x + 0y + 0z = -2
0x + y + 0z = 3
0x + 0y + z = 5
so x = -2, y = 3, z = 5
check:
x + 3y - z = 2 --> -2 + 9 - 5 = 7 - 5 = 2, yes
x - 2y + 3z = 7 --> -2 - 6 + 15 = -8 + 15 = 7, yes
x + 2y - 5z = -21 --> -2 + 6 - 25 = 4 - 25 = -21, yes