In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {5x+12y+z=102x+5y+2z=−1x+2y−3z=5\begin{cases} 5x + 12y + z = 10 \\ 2x + 5y + 2z = -1 \\ x + 2y - 3z = 5 \end{cases}⎩⎨⎧5x+12y+z=102x+5y+2z=−1x+2y−3z=5

Ch. 6 - Matrices and Determinants

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

Problem 1

Chapter 7, Problem 1

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases}\)5x + 12y + z = 10 \\2x + 5y + 2z = -1 $\x$ + 2y - 3z = 5\(\end{cases}$

Verified step by step guidance

Write the system of equations as an augmented matrix: \[\left[\begin{array}{ccc|c} 5 & 12 & 1 & 10 \\ 2 & 5 & 2 & -1 \\ 1 & 2 & -3 & 5 \end{array}\right]\]

Use row operations to create a leading 1 in the first row, first column if needed, or use the existing pivot to eliminate the x-terms in the rows below. For example, use row 3 as a pivot to eliminate x in rows 1 and 2.

Perform row operations to create zeros below the pivot in the first column. This means subtracting appropriate multiples of the first row from the second and third rows to eliminate the x-terms in those rows.

Move to the second row and second column to create a pivot (leading 1) there, then use it to eliminate the y-term in the third row by appropriate row operations.

Once the matrix is in upper triangular (row echelon) form, use back substitution to solve for the variables starting from the last row and moving upward.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Solutions can be unique, infinite, or nonexistent depending on the system's consistency and independence.

Gaussian Elimination

Gaussian elimination is a 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 Row-Echelon Form

Row operations include swapping rows, multiplying a row by a nonzero scalar, and adding multiples of one row to another. These operations are used to convert a matrix into row-echelon form, where each leading coefficient is to the right of the one above it, facilitating straightforward solution extraction.

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Related Practice

Textbook Question

Write the augmented matrix for each system of linear equations.

$\begin{cases}\)2x + y + 2z = 2 \\3x - 5y - z = 4 $\x$ - 2y - 3z = -6\(\end{cases}$

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Textbook Question

a. Give the order of each matrix.

b. If $A = [a_{i j}]$ , identify $a sub 32$ and $a sub 23$ , or explain why identification is not possible.

$\begin{bmatrix}\)4 & -7 & 5 \\-6 & 8 & -1\(\end{bmatrix}$

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Textbook Question

Perform each matrix row operation and write the new matrix.

$$\begin{bmatrix}$1 & 2 & 2 & $\vert$ & 2 \\0 & 1 & -1 & $\vert$ & 2 \\0 & 5 & 4 & $\vert$ & 1$\end{bmatrix}$-5R_2 + R_3$

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Textbook Question

Find the products AB and BA to determine whether B is the multiplicative inverse of A.

$A = $\begin{bmatrix}$4 & -3 \\-5 & 4$\end{bmatrix}$, B = $\begin{bmatrix}$4 & 3 \\5 & 4$\end{bmatrix}$$

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Textbook Question

Evaluate each determinant in Exercises 1–10.

$\begin{vmatrix}\)5 & 7 \\2 & 3\(\end{vmatrix}$

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Textbook Question

a. Give the order of each matrix.

b. If A = [aᵢⱼ] , identify a₃₂ and a₂₃, or explain why identification is not possible.

$\begin{bmatrix}\)1 & -5 & $\pi$ & e \\0 & 7 & -6 & -$\pi$ \\-2 & $\frac{1}{2}$ & 11 & -\(\frac{1}{5}\]\end{bmatrix}$

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