What is the augmented matrix of the following system?
-3x + 5y = 2
6x + 2y = 7

Verified step by step guidance

1. Identify the coefficients of the variables and the constants from each equation in the system. For the first equation, -3x + 5y = 2, the coefficients are -3 for x and 5 for y, and the constant is 2.
2. For the second equation, 6x + 2y = 7, the coefficients are 6 for x and 2 for y, and the constant is 7.
3. Write the coefficients of the variables in a matrix form, placing each equation's coefficients in a separate row. This forms the coefficient matrix:
4. \[\begin{bmatrix} -3 & 5 \\ 6 & 2 \end{bmatrix}\]
5. To form the augmented matrix, append the constants as an additional column to the coefficient matrix, resulting in:
6. \[\begin{bmatrix} -3 & 5 & 2 \\ 6 & 2 & 7 \end{bmatrix}\]

Key Concepts

System of Linear Equations

A system of linear equations 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. Understanding the structure of these equations is essential for representing and solving them using matrices.

Augmented Matrix

An augmented matrix is a compact representation of a system of linear equations. It combines the coefficient matrix and the constants from the equations into one matrix, where each row corresponds to an equation and the last column contains the constants. This form is useful for applying matrix operations to solve the system.

Matrix Representation of Equations

Matrix representation involves organizing the coefficients of variables and constants into rows and columns. For example, the coefficients of variables form the main part of the matrix, while the constants form an additional column. This representation simplifies solving systems using methods like Gaussian elimination.