Answer to Question #206888 in Algebra for shah

Question

solve the following using gaussian elimination method 2x + y + 3z = 4 x + y + 2z = 0 2x + 4y + 6z = -8

Answer 1

Write x+2y+3z=1 as a row in an Augmented Matrix:

[123∣1]

Add row for the equation 2x+5y+7z=2 :

[123∣1257∣2]

Add row for the equation 3x+5y+7z=4 :

⎡⎢⎣123∣1257∣2357∣4⎤⎥⎦

The augmented matrix is complete. Perform Elementary Row Operations.

We want the coefficient in position (1,1) to be one and it is, therefore, no operation is required.

We want the other two coefficients is column 1 to be 0, therefore, we perform the following two row operations:

R2−2R1→R2

⎡⎢⎣123∣1011∣0357∣4⎤⎥⎦

R3−3R1→R3

⎡⎢⎣123∣1011∣00−1−2∣1⎤⎥⎦

We want the coefficient in position (2,2) to be 1 and it is, therefore, no operation is required.

We want the other two coefficients in column 2 to be 0, therefore, we perform the following two row operations:

R1−2R2→R1

⎡⎢⎣101∣1011∣00−1−2∣1⎤⎥⎦

R3+R2→R3

⎡⎢⎣101∣1011∣000−1∣1⎤⎥⎦

We want the coefficient in position (3,3) to be 1 and it is, therefore, we multiply the row by -1:

−1R3→R3

⎡⎢⎣101∣1011∣0001∣−1⎤⎥⎦

We want the other two coefficients in column 3 to be 0, therefore, we perform the following two row operations:

R2−R3→R2

⎡⎢⎣101∣1010∣1001∣−1⎤⎥⎦

R1−R3→R1

⎡⎢⎣100∣2010∣1001∣−1⎤⎥⎦

We have an identity matrix on the left, therefore, the solutions are on the right:

x=2,y=1,andz=−1