Write each matrix equation as a system of linear equations without matrices.
$\(\begin{bmatrix}\)4 & -7 \\2 & -3\(\end{bmatrix}\[\begin{bmatrix}\)x \(\y\]\end{bmatrix}\)=\(\begin{bmatrix}\)-3 \\1\(\end{bmatrix}\)$

Verified step by step guidance

1. Identify the given matrix equation: \(\left[ \begin{array}{cc} 4 & -7 \\ 2 & -3 \end{array} \right] \left[ \begin{array}{c} x \\ y \end{array} \right] = \left[ \begin{array}{c} -3 \\ 1 \end{array} \right]\).
2. Recall that multiplying a matrix by a vector corresponds to forming linear combinations of the vector components with the matrix rows.
3. Write the first row multiplication as an equation: \$4x - 7y = -3$.
4. Write the second row multiplication as an equation: \$2x - 3y = 1$.
5. Thus, the matrix equation is equivalent to the system of linear equations: \(\begin{cases} 4x - 7y = -3 \\ 2x - 3y = 1 \end{cases}\).

Key Concepts

Matrix Multiplication

Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. In this problem, multiplying the 2x2 coefficient matrix by the 2x1 variable matrix results in a 2x1 matrix representing the system's left side.

System of Linear Equations

A system of linear equations consists of multiple linear equations with the same variables. Writing the matrix equation as a system means expressing each row multiplication as an individual linear equation involving variables x and y.

Equating Matrices to Form Equations

When two matrices are equal, their corresponding entries are equal. This principle allows us to set each element of the product matrix equal to the corresponding element in the constant matrix, forming a system of equations to solve.