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 $6 .$

$2 x-3 y=9$
$4 x+3 y=9$

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more equations that share common variables. The goal is to find values for the variables that satisfy all equations simultaneously, which geometrically represents the intersection point(s) of the lines (or planes in higher dimensions).

Elimination Method

The elimination method is a technique used to solve systems of linear equations by adding or subtracting scaled versions of equations to cancel out one of the variables. This reduction simplifies the system, making it straightforward to solve for the remaining variable and back-substitute to find the others.

Consistency and Uniqueness of Solutions

When solving a system of equations, it is important to determine whether the system is consistent or inconsistent. A consistent system has at least one solution; if there is exactly one solution, the system is independent and uniquely determined, while infinitely many solutions indicate a dependent system. An inconsistent system has no solution because the equations contradict each other.

Ordered-Pair Notation

Ordered-pair notation is used to represent the solution of a system with two variables. The solution is written in the form (x, y), where 'x' is the value of the first variable and 'y' is the value of the second variable that together satisfy all the given equations in the system.

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