In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {3x+4y+2z=34x−2y−8z=−4x+y−z=3\begin{cases} 3x + 4y + 2z = 3 \\ 4x - 2y - 8z = -4 \\ x + y - z = 3 \end{cases}⎩⎨⎧3x+4y+2z=34x−2y−8z=−4x+y−z=3

Ch. 6 - Matrices and Determinants

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

문제 5

7장, 문제 5

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

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Write the system of equations as an augmented matrix: \[\left[\begin{array}{ccc|c} 3 & 4 & 2 & 3 \\ 4 & -2 & -8 & -4 \\ 1 & 1 & -1 & 3 \end{array}\right]\]

Use row operations to create a leading 1 in the first row, first column. For example, swap rows if needed or divide the first row by the coefficient of $x$ in the first equation.

Eliminate the $x$-terms from the second and third rows by replacing those rows with suitable linear combinations of the first row and themselves, aiming to get zeros below the leading 1 in the first column.

Move to the second row and create a leading 1 in the second column (pivot position). Then eliminate the $y$-term from the third row by using row operations.

Finally, use back substitution starting from the last row to express $z$, then substitute back to find $y$, and then $x$, thus finding the complete solution to the system.

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12m

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 these 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 by 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.