. Solve by Gauss-Elimination method 3x+4y+5z= 18
2x-y+8z=13
5x-26-7z=20
Answers
Answer:
Hence the solution is x=3 y=1 z=1
hope this helps you ^^^
Answer:
To solve the system of equations using the Gauss-Elimination method, we'll perform row operations to transform the augmented matrix into row-echelon form. Here are the steps:
Step 1: Write the augmented matrix for the given system of equations:
[ 3 4 5 | 18 ]
[ 2 -1 8 | 13 ]
[ 5 -26 -7 | 20 ]
Step 2: Perform row operations to eliminate the coefficients below the first entry in the first column (3):
R2 = R2 - (2/3)R1
R3 = R3 - (5/3)R1
The new matrix becomes:
[ 3 4 5 | 18 ]
[ 0 -11/3 26/3 | 1 ]
[ 0 -38/3 -38/3 | -14/3 ]
Step 3: Now, eliminate the coefficient below the second entry in the second column (-11/3):
R3 = R3 - (-11/3)R2
The new matrix becomes:
[ 3 4 5 | 18 ]
[ 0 -11/3 26/3 | 1 ]
[ 0 0 -16/3 | -17/3 ]
Step 4: Solve for the variables using back-substitution.
From the last row of the matrix:
(-16/3)z = -17/3
Simplifying, we get:
z = (-17/3) / (-16/3)
z = 17/16
Substituting z = 17/16 back into the second row:
(-11/3)y + (26/3)(17/16) = 1
(-11/3)y + 221/48 = 1
(-11/3)y = 1 - 221/48
(-11/3)y = (48 - 221)/48
(-11/3)y = -173/48
Simplifying, we get:
y = (-173/48) * (-3/11)
y = 173/176
Finally, substituting the values of y and z into the first row:
3x + 4(173/176) + 5(17/16) = 18
3x + 692/176 + 85/16 = 18
3x + 692/176 + 85/16 = 288/16
3x + 692/176 + 85/16 = 18
3x + 692/176 + 85/16 = 18
3x + 692/176 + 85/16 = 18
3x + 692/176 + 85/16 = 18
3x = (288/16) - (692/176) - (85/16)
3x = (576 - 389 - 85)/16
3x = 102/16
3x = 51/8
x = (51/8) * (1/3)
x = 51/24
x = 17/8
Therefore, the solution to the system of equations is:
x = 17/8, y = 173/176, z = 17/16.
Step-by-step explanation: