The solution to the system of equations x+y=1, 2x-y+z=1, and x+2y+z=8/3 is x=7/6, y=-1/6, and z=-1/2, obtained through substitution.
Here's how you can solve the system:
Method 1: Substitution
Solve the first equation for x: x = 1 - y
Substitute this expression for x in the second and third equations:
2(1 - y) - y + z = 1
(1 - y) + 2y + z = 8/3
Solve the resulting equations for y and z:
y = -1/6
z = -1/2
Substitute the values for y and z back into the first equation to find x:
x + (-1/6) = 1
x = 7/6
Method 2: Elimination
Add the first and second equations to eliminate x:
3y + z = 2
Solve the third equation for z and substitute it into the equation above:
3y - 3/2 = 2
y = -1/6
Substitute the value for y back into the equation from step 1 to find z:
3(-1/6) + z = 2
z = -1/2
Substitute the values for y and z back into the first equation to find x:
x + (-1/6) = 1
x = 7/6
Both methods lead to the same solution: x = 7/6, y = -1/6, and z = -1/2.