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. 6x + 5y = 13 5x + 4y = 10

Verified step by step guidance

1. Identify the variables and write the system of equations clearly: 6x + 5y = 13 and 5x + 4y = 10.
2. Extract the coefficients of the variables x and y from each equation to form the coefficient matrix A. For the first equation, the coefficients are 6 and 5; for the second, 5 and 4. So, A = \( \begin{bmatrix} 6 & 5 \\ 5 & 4 \end{bmatrix} \).
3. Form the variable matrix X, which contains the variables x and y as a column matrix: X = \( \begin{bmatrix} x \\ y \end{bmatrix} \).
4. Form the constant matrix B, which contains the constants from the right side of the equations: B = \( \begin{bmatrix} 13 \\ 10 \end{bmatrix} \).
5. Write the matrix equation in the form AX = B, which represents the system of equations as \( \begin{bmatrix} 6 & 5 \\ 5 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 13 \\ 10 \end{bmatrix} \).

Key Concepts

Linear Systems of Equations

A linear system consists of two or more linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. In this problem, the system has two equations with two variables, x and y.

Matrix Representation of Linear Systems

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

Coefficient and Constant Matrices

The coefficient matrix A is formed by the coefficients of variables from each equation, arranged in rows corresponding to each equation. The constant matrix B contains the constants from the right side of the equations. For example, A = [[6,5],[5,4]] and B = [[13],[10]] for this system.