Solve using the elimination method. If a system has an...

Solve using the elimination method. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If a system has no solution, state this.
$\begin{aligned} x+y &=6 \\-x+2 y &=15 \end{aligned}$

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more equations with the same set of variables. The solution of the system is the set of values for the variables that satisfies every equation simultaneously. Understanding these systems is fundamental in algebra because they model situations where multiple conditions must be met at once.

Elimination Method

The elimination method is a technique used to solve systems of equations by combining the equations in such a way as to eliminate one of the variables. This is achieved by adding or subtracting the equations after appropriate scaling, allowing you to isolate one variable and solve for it, thereby reducing the system's complexity. The method is particularly useful when the coefficients of one variable are opposites.

Types of Solutions for Systems

Systems of linear equations can have a unique solution, infinitely many solutions, or no solution. A unique solution occurs when the equations intersect at a single point, infinitely many solutions occur when the equations represent the same line (making them dependent), and no solution happens when the equations represent parallel lines (making the system inconsistent). Recognizing the type of solution is essential for properly interpreting the results of the elimination process.

Set-Builder Notation

Set-builder notation is a mathematical shorthand used to describe a set by specifying a property or rule that its members must satisfy. In the context of systems of equations that have infinitely many solutions, set-builder notation is often used to describe the entire solution set by expressing one variable in terms of a parameter. This concise form is valuable in clearly communicating the infinite nature of the solution set.

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