Question 1002469: 3x+2y-8z=29
9x-y+2z=23
-x-2y+8z=-11
Found 3 solutions by mananth, ikleyn, n2:
Answer by mananth(16949)   (Show Source):
You can put this solution on YOUR website! 3 x + 2 y + -8 z = 29 -------------- 1
9 x + -1 y 2 z = 23 -------------- 2
-1 x + -2 y + 8 z -11 -------------- 3
consider equation 1 &2 Eliminate y
Multiply 1 by 1 -5
Multiply 2 by 2 4
we get
3 x + 2 y + -8 z = 29
18 x + -2 y + 4 z = 46
Add the two
21 x + 0 y + -4 z = 75 ------------- 4
consider equation 2 & 3 Eliminate y
Multiply 2 by -2
Multiply 3 by 1
we get
-18 x + 2 y + -4 z = -46
-1 x + -2 y + 8 z = -11
Add the two
-19 x + 0 y + 4 z = -57 -------------5 5
Consider (4) & (5) Eliminate x
Multiply 4 by 19
Multiply (5) by 21
we get
399 x + -76 z = 1425
-399 x + 84 z = -1197
Add the two
0 x + 8 z = 228
/ 8
z = 28.50
Plug the value of z in (5)
-19 x + 4 z = -57
-19 x = -171
x = 9
plug value of x & z in 1
27 + 2 y + -228 = 29
2 y = 29 + -27 + 228
y = 115
x= 9.00 ,y= 115.00 ,z= 28.50
Answer by ikleyn(53746)  (Show Source):
You can put this solution on YOUR website! .
3x + 2y - 8z = 29
9x - y + 2z = 23
-x - 2y + 8z = -11
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
When an assignment is to solve 3x3-system of equations, the goal is usually to teach a student to something useful.
For example, to teach to look at equations and to search for something that makes the solution easy and educative.
As usual (if the creators of the assignment are professionals), the goal is not simply manipulate with numbers,
but to find some useful pattern.
In this system, the first and the third equations have similar parts "2y - 8z" and "-2y + 8z",
so if we add these equations, we eliminate 'y' and 'z' simultaneously and easy will find 'z'.
It is the key to start.
So, your original system of equation is
3x + 2y - 8z = 29 (1)
9x - y + 2z = 23 (2)
-x - 2y + 8z = -11 (3)
Add equations (1) and (3). The terms with 'y' and 'z' will cancel each other, and you will get then
3x - x = 29 - 11, ---> 2x = 18, ---> x = 18/2 = 9.
Now substitute this value x = 9 into the first and second equations
3*9 + 2y - 8z = 29 (1')
9*9 - y + 2z = 23 (2')
Simplify
2y - 8z = 2 (1'')
-y + 2z = -58 (2'')
Now we are on the straight finish line: we only need to solve one 2x2-syatem of equations (1''), (2'').
Solve it by the Elimination method. For it, multiply equation (2'') by 2 (both sides) and add to equation (1'').
You will get
- 8z + 4z = 2 + 2*(-58), ---> -4z = -114, z = (-114)/(-4) = 28.5.
Now from equation (2'')
y = 58 + 2*28.5 = 115.
ANSWER. The solution to the given system is x= 9, y = 115, z = 28.5.
Solved.
I hope that you will learn something useful from my solution.
Answer by n2(78)  (Show Source):
You can put this solution on YOUR website! .
3x + 2y - 8z = 29
9x - y + 2z = 23
-x - 2y + 8z = -11
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
So, your original system of equation is
3x + 2y - 8z = 29 (1)
9x - y + 2z = 23 (2)
-x - 2y + 8z = -11 (3)
Add equations (1) and (3). The terms with 'y' and 'z' will cancel each other, and you will get then
3x - x = 29 - 11, ---> 2x = 18, ---> x = 18/2 = 9.
Now substitute this value x = 9 into the first and second equations
3*9 + 2y - 8z = 29 (1')
9*9 - y + 2z = 23 (2')
Simplify
2y - 8z = 2 (1'')
-y + 2z = -58 (2'')
Now we are on the straight finish line: we only need to solve one 2x2-syatem of equations (1''), (2'').
Solve it by the Elimination method. For it, multiply equation (2'') by 2 (both sides) and add to equation (1'').
You will get
- 8z + 4z = 2 + 2*(-58), ---> -4z = -114, z = (-114)/(-4) = 28.5.
Now from equation (2'')
y = 58 + 2*28.5 = 115.
ANSWER. The solution to the given system is x= 9, y = 115, z = 28.5.
Solved.
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