Write each matrix equation as a system of equations and...

Write each matrix equation as a system of equations and solve the system by the method of your choice.
$$\left[\begin{array}{ll}2 & 3 \\4 & 6\end{array}\right]\left[\begin{array}{l}x \\y\end{array}\right]=\left[\begin{array}{l}5 \\
9\end{array}\right]$$

Key Concepts

Matrix Equation

A matrix equation is a compact representation of a system of linear equations where the matrix contains the coefficients of the unknown variables, the variable vector symbolizes the set of unknowns, and the resulting vector represents the constants. This form is central to linear algebra because it facilitates the use of systematic methods and algorithms for solving multiple equations simultaneously.

System of Linear Equations

A system of linear equations consists of several equations that must be satisfied simultaneously by the same set of unknowns. Expressing a matrix equation as individual equations allows one to analyze the relationships between the variables and to determine whether the system has a unique solution, infinitely many solutions, or no solution at all.

Methods for Solving Systems

Common methods for solving systems, such as substitution and elimination, involve manipulating the equations to reduce the number of unknowns step by step. These methods are particularly useful in transforming and simplifying the system, thereby making it easier to identify the solution set or to conclude that the system is inconsistent.

Inconsistency in Linear Systems

An inconsistent system arises when the equations impose contradictory constraints on the unknowns, such that no single set of values can satisfy all the equations simultaneously. Detecting inconsistency is a critical part of the analysis since it indicates that the system has no solution, which can often be revealed by observing proportional rows in the coefficient matrix that correspond to non-proportional entries in the constant vector.

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