In Exercises 33 - 36, write each matrix equation as a system of linear equations without matrices.

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1. Identify the matrix equation given: $\left[\begin{array}{cc}4& -7\\ 2& -3\end{array}\right]\left[x\right]=[-3]$ and $\left[y\right]=\left[1\right]$ combined as $\left[\begin{array}{cc}4& -7\\ 2& -3\end{array}\right][x,y]=[-3,1]$.
2. Recall that multiplying a matrix by a vector corresponds to forming linear combinations of the vector components with the matrix entries. So, the first row of the matrix times the vector $x,y$ equals the first entry on the right side, and the second row times the vector equals the second entry.
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 rewritten as the system of linear equations: $\backslash begin\{cases\}\; 4x\; -\; 7y\; =\; -3\; \backslash \backslash \; 2x\; -\; 3y\; =\; 1\; \backslash end\{cases\}$.

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 outputs.

System of Linear Equations

A system of linear equations consists of multiple linear equations with the same variables. Each row in the matrix equation corresponds to one linear equation, which can be written by equating the sum of products to the corresponding constant.

Translating Matrix Equations to Linear Equations

To convert a matrix equation to a system of linear equations, multiply each row of the coefficient matrix by the variable vector and set the result equal to the corresponding entry in the constant vector. This process reveals the individual linear equations.