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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Identify the coefficients of each variable in the system of equations.
Write the coefficients of the variables and the constants as rows in the matrix.
The first row corresponds to the equation: $4x - 2y + 3z - 4 = 0$, so it becomes: $[4, -2, 3, -4]$.
The second row corresponds to the equation: $3x + 5y + z - 7 = 0$, so it becomes: $[3, 5, 1, -7]$.
The third row corresponds to the equation: $5x - y + 4z - 7 = 0$, so it becomes: $[5, -1, 4, -7]$.
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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 is a matrix that represents a system of linear equations, where each row corresponds to an equation and each column corresponds to the coefficients of the variables, along with an additional column for the constants on the right side of the equations. For example, the system of equations can be transformed into a matrix format that simplifies the process of analyzing the system without solving it.
The dimension of a matrix refers to its size, expressed in terms of the number of rows and columns it contains. For an augmented matrix, the dimension is typically given as 'm x n', where 'm' is the number of equations (rows) and 'n' is the number of variables plus one (for the augmented part). Understanding the dimension helps in determining the nature of the solutions to the system.
Linear equations are mathematical statements that express a relationship between variables in which each term is either a constant or the product of a constant and a single variable. In the context of the given problem, each equation represents a plane in three-dimensional space, and the solution to the system corresponds to the intersection of these planes, which can be visualized geometrically.