In Exercises 49–52, use matrix multiplication to determine...

In Exercises 49–52, use matrix multiplication to determine whether each matrix is a solution of the system of equations. Use a graphing utility to verify your results. $$ \begin{array}{l}{\left\{\begin{array}{c}{x+2 y=4} \\ {3 x+2 y=0}\end{array}\right.} \\ {\begin{array}{ll}{\text { (a) }\left[\begin{array}{c}{2} \\ {1}\end{array}\right]} & {\text { (b) }\left[\begin{array}{r}{-2} \\ {3}\end{array}\right]} \\ {\text { (c) }\left[\begin{array}{r}{-4} \\ {4}\end{array}\right]} & {\text { (d) }\left[\begin{array}{r}{2} \\ {-3}\end{array}\right]}\end{array}}\end{array} $$

In Exercises 49–52, use matrix multiplication to determine whether each matrix is a solution of the system of equations. Use a graphing utility to verify your results.
$$
\begin{array}{l}{\left\{\begin{array}{c}{x+2 y=4} \\ {3 x+2 y=0}\end{array}\right.} \\ {\begin{array}{ll}{\text { (a) }\left[\begin{array}{c}{2} \\ {1}\end{array}\right]} & {\text { (b) }\left[\begin{array}{r}{-2} \\ {3}\end{array}\right]} \\ {\text { (c) }\left[\begin{array}{r}{-4} \\ {4}\end{array}\right]} & {\text { (d) }\left[\begin{array}{r}{2} \\ {-3}\end{array}\right]}\end{array}}\end{array}
$$

Key Concepts

Graphical Verification

Graphical verification uses graphing utilities to plot the individual equations of a system. The intersection point(s) of these graphs indicate the solution(s) of the system. This visual method provides an intuitive check for the results obtained through algebraic methods, ensuring consistency between analytical and graphical approaches.

Solution Verification via Substitution

Verifying a potential solution to a system of equations often involves substituting the candidate values into the original equations or the associated matrix equation. If the resulting products align with the constant terms or the other side of the equation, the candidate is a valid solution. This direct checking method is essential for confirming or rejecting proposed solutions.

Systems of Linear Equations

Systems of linear equations consist of two or more linear equations that share common variables. Representing such systems in a matrix form simplifies many operations including solution verification and solving. Understanding this concept is fundamental to many topics in linear algebra and applied mathematics.

Matrix Multiplication

Matrix multiplication involves computing the dot product of rows of one matrix with columns of another. In the context of linear systems, it is used to multiply the coefficient matrix by a candidate solution vector to verify whether the resulting vector matches the constant terms. This method offers a systematic way to check the validity of potential solutions.

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