Write the augmented matrix for each system and give its dimension. Do not solve. 4x - 2y + 3z - 4 = 0 3x + 5y + z - 7 = 0 5x - y + 4z - 7 = 0

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Rewrite each equation in the form where all variables and constants are separated, moving the constant term to the right side. For example, the first equation becomes $4x - 2y + 3z = 4$.

Identify the coefficients of the variables $x$, $y$, and $z$ in each equation and the constant term on the right side. For the first equation, the coefficients are 4, -2, 3 and the constant is 4.

Construct the augmented matrix by placing the coefficients of $x$, $y$, and $z$ in the first three columns and the constants in the last column. Each row corresponds to one equation.

Write the augmented matrix as: \[\left[\begin{array}{ccc|c} 4 & -2 & 3 & 4 \\ 3 & 5 & 1 & 7 \\ 5 & -1 & 4 & 7 \end{array}\right]\]

Determine the dimension of the augmented matrix by counting the number of rows and columns. Here, the matrix has 3 rows (equations) and 4 columns (3 variables plus 1 constant column), so its dimension is $3 \times 4$.

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Key Concepts

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

Augmented Matrix

An augmented matrix represents a system of linear equations by combining the coefficient matrix and the constants into one matrix. Each row corresponds to an equation, and each column corresponds to a variable or the constants. This format simplifies the process of applying matrix operations to solve or analyze the system.

Matrix Dimensions

The dimension of a matrix is given by the number of rows and columns it contains, expressed as 'rows × columns'. For an augmented matrix, the rows equal the number of equations, and the columns equal the number of variables plus one for the constants. Understanding dimensions is essential for matrix operations and system analysis.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. Each equation represents a linear relationship, and the goal is often to find values of variables that satisfy all equations simultaneously. Writing the system in matrix form helps in organizing and solving these equations efficiently.

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