21-50= Solving a System of Equations Solve the system, or...

$21-50=$ Solving a System of Equations Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example $6 .$

$x+3 y=5$
$2 x-y=3$

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more equations with the same set of variables. The goal is to find values for these variables that satisfy all the equations simultaneously. This concept is fundamental in algebra and appears in various applications in mathematics and other fields.

Elimination Method

The elimination method is a technique for solving systems of linear equations by adding or subtracting equations to eliminate one of the variables. This reduction simplifies the system into a single equation with one unknown, which can then be solved easily before back-substituting to determine the remaining variable(s).

Substitution Method

The substitution method involves solving one of the equations for one variable and then substituting this expression into the other equation(s). This approach reduces the number of variables in the system and is particularly useful when one equation can be easily manipulated to isolate a variable.

Consistency and Uniqueness of Solutions

Systems of linear equations can be classified based on their solutions; they may have a unique solution (consistent and independent), infinitely many solutions (consistent and dependent), or no solution (inconsistent). Understanding these classifications helps in determining the nature of the system and the appropriate method for solving it.

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