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 $6 .$

$\left\{\begin{aligned} 3 x+2 y &=8 \\ x-2 y &=0 \end{aligned}\right.$

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Key Concepts

Consistency and Types of Solutions

Systems of linear equations can be categorized as consistent or inconsistent. A consistent system has at least one solution and can be further classified as having a unique solution or infinitely many solutions (dependent system). In contrast, an inconsistent system has no solution, often because the equations represent parallel lines that never intersect.

Elimination Method

The elimination method involves combining the equations in a system to cancel out one of the variables. By aligning the coefficients (sometimes through multiplication), the equations can be added or subtracted to eliminate a variable, leaving an equation that can be solved directly.

Substitution Method

The substitution method is a technique for solving systems of equations where one equation is solved for one variable, and that expression is substituted into the other equations. This process reduces the system to a single equation with one variable, making it easier to solve.

System of Linear Equations

A system of linear equations consists of two or more equations that involve the same set of variables. The goal is to find the values of these variables that satisfy all the equations simultaneously, which is fundamental in algebra and applicable to various real-world scenarios.

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