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 $6 .$

$\left\{\begin{aligned} 2 x-3 y &=-8 \\ 14 x-21 y &=3 \end{aligned}\right.$

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Key Concepts

Consistency and Inconsistency in Systems

A system of equations is said to be consistent if there exists at least one solution; it is inconsistent if no solution exists. In the context of linear systems, inconsistency often arises when the equations represent parallel lines that never intersect.

Scalar Multiplication and Proportionality

Scalar multiplication involves multiplying an equation by a constant to produce an equivalent equation. If one equation in the system is a scalar multiple of another, the equations are either identical (leading to infinitely many solutions) or contradictory if the constants do not align properly, resulting in no solution.

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 a common solution that satisfies all the equations simultaneously, which can result in a unique solution, infinitely many solutions, or no solution at all.

Elimination Method

The elimination method is a systematic approach to solving systems of linear equations by adding or subtracting equations to eliminate one of the variables. This method simplifies the system into a single equation with one variable, making it easier to solve for the remaining variables.

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