In Exercises 29 - 32, write each linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix. x + 3y + 4z = - 3 x + 2y + 3z = - 2 x + 4y + 3z = - 6

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1. Identify the variables and write the system of equations clearly: x + 3y + 4z = -3, x + 2y + 3z = -2, x + 4y + 3z = -6.
2. Extract the coefficients of the variables x, y, and z from each equation to form the coefficient matrix A. This matrix contains only the numbers multiplying the variables: A =     In MathML, this is: $\begin{array}{ccc}1& 3& 4\\ 1& 2& 3\\ 1& 4& 3\end{array}$
3. Form the variable matrix X, which is a column matrix containing the variables x, y, and z: $\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$
4. Form the constant matrix B, which is a column matrix containing the constants from the right side of the equations: $\left[\begin{array}{c}-3\\ -2\\ -6\end{array}\right]$
5. Write the matrix equation in the form AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

Key Concepts

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 solve these systems is fundamental in algebra.

Matrix Representation of Linear Systems

A system of linear equations can be expressed compactly as a matrix equation AX = B, where A is the coefficient matrix containing the coefficients of variables, X is the column matrix of variables, and B is the constant matrix. This form simplifies manipulation and solution of the system.

Coefficient and Constant Matrices

The coefficient matrix A includes all coefficients of the variables arranged by equation and variable order, while the constant matrix B contains the constants from the right side of each equation. Correctly identifying these matrices is essential for forming the matrix equation.