Answer:
To solve the system of equations using Gaussian elimination, we'll perform row operations to eliminate the variable x. The given system of equations is:
x + 2y - z = 2
-x + 3y + 2z = 13
2x - y + 3z = 19
First, let's use equations 1 and 2 to eliminate x. Add equation 1 to equation 2:
(1) + (2): (x + 2y - z) + (-x + 3y + 2z) = 2 + 13
Simplifying, we get:
5y + z = 15 ...........(4)
Now, we have two equations with variables y and z:
5y + z = 15
2x - y + 3z = 19
Next, we'll use equations 3 and 4 to eliminate x. Multiply equation 3 by 2 and equation 4 by 5:
2 * (3): 4x - 2y + 6z = 38
5 * (4): 25y + 5z = 75
Now, let's add the two equations:
(2 * (3)) + (5 * (4)): (4x - 2y + 6z) + (25y + 5z) = 38 + 75
Simplifying, we get:
4x + 23y + 11z = 113 ...........(5)
Now we have three equations with variables y and z:
5y + z = 15
4x + 23y + 11z = 113
To eliminate x, we'll multiply equation 4 by 4:
4 * (4): 20y + 4z = 60
Now, let's subtract equation 5 from this new equation:
(4 * (4)) - (5): (20y + 4z) - (4x + 23y + 11z) = 60 - 113
Simplifying, we get:
-4x - 3y - 7z = -53 ...........(6)
Now we have two equations with variables y and z:
-4x - 3y - 7z = -53
5y + z = 15
We can solve this system of equations using Gaussian elimination. Let's eliminate y by multiplying equation 6 by 5 and equation 4 by 3:
5 * (6): -20x - 15y - 35z = -265
3 * (4): 15y + 3z = 45
Now, add the two equations:
(5 * (6)) + (3 * (4)): (-20x - 15y - 35z) + (15y + 3z) = -265 + 45
Simplifying, we get:
-20x - 32z = -220 ...........(7)
Now we have two equations with variables x and z:
-20x - 32z = -220
5y + z = 15
To eliminate z, we'll multiply equation 7 by 5 and equation 4 by 32:
5 * (7): -100x - 160z = -1100
32 * (4): 160y + 32z = 480
Now, add the two equations:
(5 * (7)) + (32 * (4)): (-100x - 160z