Answer:
(x, y, z) = (2, -1, 5)
Explanation:
You want the solution to the system of equations ...
- 2x +y +z = 8
- x +2y -z = -5
- z = 2x -y
Solution
The last equation gives an expression for z that can be substituted into the first two equations.
2x +y +(2x -y) = 8 ⇒ 4x = 8
x +2y -(2x -y) = -5 ⇒ -x +3y = -5
The first of these reduced equations is easily solve for x:
x = 8/4 = 2
Using that in the second of the reduced equations, we have ...
-2 +3y = -5
3y = -3 . . . . . . . . add 2
y = -1 . . . . . . . divide by 3
Now, we can find z:
z = 2x -y = 2(2) -(-1) = 5
The solution is (x, y, z) = (2, -1, 5).
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Additional comment
While we can always write the system of equations as an augmented matrix and ask a calculator to solve it (see attachment), it can be just as easy to take a look at the equations and develop a strategy. The given expression for z (3rd equation) makes substitution a good solution strategy. As it happens, x is developed right away by substituting for z in the first equation.
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