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 each variable and the constants in the system of equations.
2. For the first equation \(-3x + 5y = 2\), the coefficients are \(-3\) for \(x\), \(5\) for \(y\), and the constant is \(2\).
3. For the second equation \(6x + 2y = 7\), the coefficients are \(6\) for \(x\), \(2\) for \(y\), and the constant is \(7\).
4. Write the augmented matrix by placing the coefficients and constants into a matrix form: \(\begin{bmatrix} -3 & 5 & | & 2 \\ 6 & 2 & | & 7 \end{bmatrix}\).
5. The vertical line in the matrix separates the coefficients of the variables from the constants on the right side of the equations.

Key Concepts

Augmented Matrix

An augmented matrix is a matrix that represents a system of linear equations. It combines the coefficients of the variables and the constants from the equations into a single matrix. For example, the system -3x + 5y = 2 and 6x + 2y = 7 can be represented as an augmented matrix by placing the coefficients of x and y in the first two columns and the constants in the last column.

Linear Equations

Linear equations are mathematical statements that express a relationship between variables using a linear function. They can be written in the form Ax + By = C, where A, B, and C are constants. Understanding how to manipulate and represent these equations is crucial for forming the corresponding augmented matrix and solving the system.

Row Operations

Row operations are techniques used to manipulate the rows of a matrix to simplify it, particularly in the context of solving systems of equations. The three primary row operations are swapping two rows, multiplying a row by a non-zero scalar, and adding or subtracting rows. These operations are essential for transforming the augmented matrix into a form that makes it easier to find solutions to the system.