Final answer:
To solve the system of equations for x, y, and z, you add and subtract the given equations to isolate and solve for each variable sequentially, resulting in x = (b + c) / 2, y = (a + c) / 2, and z = (a + b) / 2.
Step-by-step explanation:
To solve for x, y, and z in terms of a, b, and c, we have the system of equations:
-
- -x + y + z = a
-
- x - y + z = b
-
- x + y - z = c
Add the first and second equations to eliminate y:
-
- -x + y + z + x - y + z = a + b
-
- 2z = a + b
-
- z = (a + b) / 2
Substitute the value of z into the third equation:
-
- x + y - (a + b) / 2 = c
-
- x + y = c + (a + b) / 2
Now we can add the first and third equations to eliminate z:
-
- -x + y + z + x + y - z = a + c
-
- 2y = a + c
-
- y = (a + c) / 2
Finally, solve for x by substituting the values of y and z into the second or first equation:
-
- x - (a + c) / 2 + (a + b) / 2 = b
-
- x = b + (a + c) / 2 - (a + b) / 2
-
- x = (b + c) / 2
In summary, the solution is:
-
- x = (b + c) / 2
-
- y = (a + c) / 2
-
- z = (a + b) / 2