Exercise 3.19 Solve each of the following systems of linear equations using Gaussian Elimination Methods. x+y=5 a. x-2y=-4 x+y+z=3 b. x-2y+3z=1 2x+y-z=2 x-y-z=0 C. 2x+y+2z=3

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Answer 1

Let's solve each system of linear equations using Gaussian Elimination.a.

Step 1: Write the augmented matrix for the system of equations.

Step 2: Perform row operations to obtain a leading 1 in the first column of the first row.

Step 3: Perform row operations to obtain a leading 1 in the second column of the second row.

Step 4: Perform row operations to obtain a leading 1 in the first column of the second row.

Step 5: Interpret the final augmented matrix to find the solution.The solution is

and

.b.

Step 1: Write the augmented matrix for the system of equations.

Step 2: Perform row operations to obtain a leading 1 in the first column of the first row.

Step 3: Perform row operations to obtain a leading 1 in the second column of the second row.

Step 4: Perform row operations to obtain a leading 1 in the third column of the third row.

Step 5: Perform row operations to obtain a leading 1 in the third column of the third row.

Step 6: Perform row operations to obtain a leading 1 in the first column of the first row.

Step 7: Perform row operations to obtain a leading 1 in the second column of the second row.

Step 8: Interpret the final augmented matrix to find the solution.The solution is

,

, and

.c.

Step 1: Write the augmented matrix for the system of equations.\[\begin{bmatrix}2 & 1 & -1 & | & 3 \\1 & -1 & -1 & | & 0 \\2 & 1 & 2 & | & 3\end{

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