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 $6 .$

$\left\{\begin{aligned} x+y &=4 \\-x+y &=0 \end{aligned}\right.$

Key Concepts

System of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. The goal is to find all values of the variables that satisfy every equation in the system simultaneously. Such systems can have a unique solution, no solution, or infinitely many solutions depending on the relationship between the equations.

Solving Methods (Elimination and Substitution)

The elimination and substitution methods are standard techniques used to solve systems of equations. The substitution method involves expressing one variable in terms of the others and substituting back into the other equation(s), while the elimination method involves adding or subtracting equations to remove one of the variables, thereby simplifying the system into a single equation with one unknown.

System Consistency and Dependencies

Systems of equations can be classified based on their solution sets as consistent (having at least one solution) or inconsistent (having no solutions). When a consistent system has exactly one solution, it is independent; if it has infinitely many solutions, it is known as dependent, meaning that there is a direct relationship between the equations.

Ordered-Pair Representation

The ordered-pair representation is a concise way to express the solution of a system involving two variables, typically in the form (x, y). This notation clearly indicates the unique or parametric solution values that satisfy all equations of the system.

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