Question 904088: Hi would you please help me with this question and could you show me the steps on how to solve it please.
Solve the system of equations:
x+y+z=1
2x+4y+2z=-4
-x+7y-3z=-35
My professor said the solution set is {(-1,-3,5)}
Can you kindly show me how he got those answers please. Thanks!
Found 2 solutions by richwmiller, jim_thompson5910:
Answer by richwmiller(17219)   (Show Source):
You can put this solution on YOUR website! 1,1,1,1
2,4,2,-4
-1,7,-3,-35
add down (-2) *row 1 to row 2
1,1,1,1
0,2,0,-6
-1,7,-3,-35
add down (1) *row 1 to row 3
1,1,1,1
0,2,0,-6
0,8,-2,-34
divide row 2 by 2
1,1,1,1
0,1,0,-3
0,8,-2,-34
add down (-8) *row 2 to row 3
1,1,1,1
0,1,0,-3
0,0,-2,-10
divide row 3 by -2
1,1,1,1
0,1,0,-3
0,0,1,5
We now have the value for the last variable.
We will work our way up and get the other solutions.
add up (0) *row 3 to row 2
1,1,1,1
0,1,0,-3
0,0,1,5
add up (-1) *row 3 to row 1
1,1,0,-4
0,1,0,-3
0,0,1,5
add up (-1) *row 2 to row 1
1,0,0,-1
0,1,0,-3
0,0,1,5
final
1,0,0,-1
0,1,0,-3
0,0,1,5
"-1","-3","5"
(-1,-3,5)
Answer by jim_thompson5910(35256)  (Show Source):
You can put this solution on YOUR website! If you solve x+y+z=1 for z, then you get
x+y+z=1
y+z=1-x
z = 1-x-y
--------------------
Plug this into the second equation
2x+4y+2z=-4
2x+4y+2(1-x-y)=-4
2x+4y+2-2x-2y=-4
2y+2 = -4
2y = -4-2
2y = -6
y = -6/2
y = -3
Because z = 1-x-y, and y = -3, we know that z = 1-x-y turns into z = 1-x-(-3) and that simplifies to z = 4-x
------------------
Now turn to the third equation -x+7y-3z=-35
-x+7y-3z=-35
-x+7y-3(4-x)=-35 ... plug in z = 4-x
-x+7y-12+3x=-35
-x+7(-3)-12+3x=-35 ... plug in y = -3, now solve for x
-x-21-12+3x=-35
2x-33 = -35
2x = -35+33
2x = -2
x = -2/2
x = -1
---------------
z = 4 - x
z = 4 - (-1)
z = 4 + 1
z = 5
==========================================================
Summary:
x = -1
y = -3
z = 5
The solution as an ordered triple is (-1,-3,5)
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