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

Unique Solution

A unique solution occurs when a system of linear equations has exactly one set of values that satisfies all equations simultaneously. This indicates that the lines represented by the equations intersect at exactly one point in the coordinate plane, reflecting consistent and independent equations.

Ordered Pair Representation

When solving a system of two-variable linear equations, the solution is typically expressed as an ordered pair (x, y). This notation succinctly represents the unique point where the two lines meet in the coordinate plane and is widely used to communicate the solution of the system.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find the set of values for these variables that satisfy all the equations simultaneously. Understanding this concept is fundamental to solving problems across algebra, where the relationships between the variables are expressed in a linear form.

Elimination Method

The elimination method is a technique used to solve systems of linear equations. It involves adding or subtracting the equations in order to eliminate one of the variables, which simplifies the system into a single-variable equation. This method is especially useful when the coefficients of one variable are opposites or can easily be made opposites.

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