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 $6 .$

$6 x+4 y=12$
$9 x+6 y=18$

Key Concepts

Parametrization of Infinite Solutions

When a system of equations has infinitely many solutions, it indicates that the equations are dependent, meaning one is a multiple of the other. The solution set can be described in parametric form by expressing the variables in terms of a free parameter, which represents the infinite number of solutions in a structured way.

Elimination and Substitution Methods

These are standard techniques for solving systems of equations. The substitution method involves solving one equation for one variable and replacing it into another equation, while the elimination method focuses on adding or subtracting equations to cancel out a variable. Both methods simplify the process of finding the common solution values.

Consistency, Inconsistency, and Dependency

Systems of equations can have a unique solution (consistent and independent), no solution (inconsistent), or infinitely many solutions (consistent and dependent). Recognizing these conditions is essential in determining how the equations behave relative to each other and in choosing an appropriate method to solve them.

Systems of Linear Equations

This concept involves dealing with two or more equations that share common variables. The goal is to find values for these variables that satisfy all the equations simultaneously. It forms a foundation in algebra and helps in understanding relations between variables in various contexts.

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