In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {8x+5y+11z=30−x−4y+2z=32x−y+5z=12\begin{cases} 8x + 5y + 11z = 30 \\ -x - 4y + 2z = 3 \\ 2x - y + 5z = 12 \end{cases}⎩⎨⎧8x+5y+11z=30−x−4y+2z=32x−y+5z=12

Ch. 6 - Matrices and Determinants

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

문제 7

7장, 문제 7

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases}\) 8x + 5y + 11z = 30 \\ -x - 4y + 2z = 3 \\ 2x - y + 5z = 12 \(\end{cases}$

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Write the system of equations as an augmented matrix: \[\begin{bmatrix} 8 & 5 & 11 & | & 30 \\ -1 & -4 & 2 & | & 3 \\ 2 & -1 & 5 & | & 12 \end{bmatrix}\]

Use row operations to create a leading 1 in the first row, first column if possible, or proceed to eliminate the x-terms in the second and third rows by adding suitable multiples of the first row to those rows.

Perform row operations to eliminate the x-term from the second and third rows, aiming to get zeros below the leading coefficient in the first column.

Next, focus on the second row to create a leading 1 in the y-position, then use it to eliminate the y-term in the third row.

Finally, use back substitution starting from the last row to express variables in terms of constants or parameters, thereby finding the complete solution or determining if no solution exists.

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주요 개념

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Systems of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to represent and interpret these systems is fundamental before applying solution methods.

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations by transforming the system's augmented matrix into row-echelon form using row operations. This process simplifies the system, making it easier to solve through back-substitution or to determine if no solution exists.

Row Operations and Consistency

Row operations include swapping rows, multiplying a row by a nonzero scalar, and adding multiples of one row to another. These operations preserve the solution set and help identify if the system is consistent (has at least one solution) or inconsistent (no solution). Recognizing inconsistent rows is key to concluding no solution exists.