Solve the system of three equations:
7x-y+z=25
2x+2y-3z=-8
x-3y+2z=9
To solve this system of equations, we can use the method of substitution or elimination.
Let's start with the method of substitution.
From the first equation, we can isolate x:
7x - y + z = 25
7x = y - z + 25
x = (y - z + 25)/7
Now, substitute the expression for x in the other two equations:
2x + 2y - 3z = -8
2((y - z + 25)/7) + 2y - 3z = -8
(2y - 2z + 50)/7 + 2y - 3z = -8
(2y - 2z + 50) + 14y - 21z = -56
16y - 23z = -106 ... (Equation 1)
x - 3y + 2z = 9
((y - z + 25)/7) - 3y + 2z = 9
(y - z + 25) - 21y + 14z = 63
-y - 20y - 21z + 14z = 38
-21y - 7z = 38 ... (Equation 2)
Now, let's solve the equations (Equation 1) and (Equation 2) simultaneously:
Multiply (Equation 2) by 4:
-84y - 28z = 152 ... (Equation 3)
Now, add (Equation 1) and (Equation 3) together:
16y - 23z + (-84y - 28z) = -106 + 152
-68y - 51z = 46
68y + 51z = -46 ... (Equation 4)
Multiply (Equation 2) by 3:
-63y - 21z = 114 ... (Equation 5)
Now, add (Equation 4) and (Equation 5) together:
68y + 51z + (-63y - 21z) = -46 + 114
5y + 30z = 68
5y = -30z + 68
y = (-30z + 68)/5
Now, substitute the expression for y into (Equation 2):
-21((-30z + 68)/5) - 7z = 38
(-630z + 1428 - 35z)/5 - 7z = 38
(-630z + 1428 - 35z - 35z*5)/5 = 38
(-700z + 1428 - 35z*5)/5 = 38
-700z + 1428 - 35z*5 = 190
-700z + 1428 - 175z = 190
-875z = -1238
z = -1238/(-875)
z = 1.416
Now, substitute the value of z into the expression for y:
y = (-30z + 68)/5
y = (-30(1.416) + 68)/5
y = (43.68)/5
y = 8.736
Finally, substitute the values of y and z into the expression for x:
x = (y - z + 25)/7
x = (8.736 - 1.416 + 25)/7
x = (32.32)/7
x = 4.617
Therefore, the solution to the system of equations is x = 4.617, y = 8.736, and z = 1.416.