Explanation Given: The system of equations is, { 3 x − 2 y + z = 15 − x + y + 2 z = − 10 x − y − 4 z = 14 Approach: The Gaussian Elimination with Back-Substitution is as follows. Write the augmented matrix of the system of linear equations. Use elementary row operations to rewrite the augmented matrix in row-echelon form. The elementary row operations are; Interchange of two rows. Multiply a row by a nonzero constant. Add a multiple of a row to another row. Write the system of linear equations corresponding to the matrix in row-echelon form and use back-substitution to find the solution. Calculation: Given the system of equations is, { 3 x − 2 y + z = 15 − x + y + 2 z = − 10 x − y − 4 z = 14 The coefficient matrix corresponding to the given system of equations is [ 3 − 2 1 − 1 1 2 1 − 1 − 4 ] The augmented matrix is given as, [ 3 − 2 1 ⋮ 15 − 1 1 2 ⋮ − 10 1 − 1 − 4 ⋮ 14 ] Solve the matrix using the three elementary row operations as mentioned in the approach part. Perform the row operation R 2 → R 2 + 1 3 R 1 as, [ 3 − 2 1 ⋮ 15 − 1 1 2 ⋮ − 10 1 − 1 − 4 ⋮ 14 ] → [ 3 − 2 1 ⋮ 15 0 1 3 7 3 ⋮ − 5 1 − 1 − 4 ⋮ 14 ] Perform the row operation R 3 → R 3 − 1 3 R 1 as, [ 3 − 2 1 ⋮ 15 0 1 3 7 3 ⋮ − 5 1 − 1 − 4 ⋮ 14 ] → [ 3 − 2 1 ⋮ 15 0 1 3 7 3 ⋮ − 5 0 − 1 3 − 13 3 ⋮ 9 ] Perform the row operation R 3 → R 3 + 1 R 2 as, [ 3 − 2 1 ⋮ 15 0 1 3 7 3 ⋮ − 5 0 − 1 3 − 13 3 ⋮ 9 ] → [ 3 − 2 1 ⋮ 15 0 1 3 7 3 ⋮ − 5 0 0 − 2 ⋮ 4 ] Perform the row operation R 3 → − 1 2 R 3 as, [ 3 − 2 1 ⋮ 15 0 1 3 7 3 ⋮ − 5 0 0 − 2 ⋮ 4 ] → [ 3 − 2 1 ⋮ 15 0 1 3 7 3 ⋮ − 5 0 0 1 ⋮ − 2 ] Perform the row operation R 2 → R 2 − 7 3

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Chapter 8.1, Problem 64E

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The system of linear equations {3x−2y+z=15−x+y+2z=−10x−y−4z=14.

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Chapter 8.1, Problem 63E

Chapter 8.1, Problem 65E

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Ch. 8.1 - Prob. 1CP Ch. 8.1 - Prob. 2CP Ch. 8.1 - Prob. 3CP Ch. 8.1 - Prob. 4CP Ch. 8.1 - Prob. 5CP Ch. 8.1 - Prob. 6CP Ch. 8.1 - Prob. 7CP Ch. 8.1 - Prob. 8CP Ch. 8.1 - Prob. 9CP Ch. 8.1 - Prob. 1E

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The system of linear equations { 3 x − 2 y + z = 15 − x + y + 2 z = − 10 x − y − 4 z = 14 .

Solution Summary: The author explains the Gaussian Elimination with Back-Substitution approach to solving the system of linear equations.

Videos

To solve:

The system of linear equations {3x−2y+z=15−x+y+2z=−10x−y−4z=14.

Want to see the full answer?

Chapter 8 Solutions

The system of linear equations { 3 x − 2 y + z = 15 − x + y + 2 z = − 10 x − y − 4 z = 14 .

Solution Summary: The author explains the Gaussian Elimination with Back-Substitution approach to solving the system of linear equations.

Videos

To solve: The system of linear equations {3x−2y+z=15−x+y+2z=−10x−y−4z=14.

Want to see the full answer?

Chapter 8 Solutions

To solve:

The system of linear equations {3x−2y+z=15−x+y+2z=−10x−y−4z=14.