Solve using the elimination method. Also determine whether...

Solve using the elimination method. Also determine whether each system is consistent or inconsistent and whether the equations are dependent or independent.
$$
\begin{aligned}
& 7(x-y)=14 \\
& 2 x=y+5
\end{aligned}
$$

Key Concepts

Dependent vs Independent Equations

Dependent equations in a system are essentially multiples of one another, representing the same line, which results in infinitely many solutions. Independent equations do not overlap in this way, typically leading to a unique solution when the system is consistent. Recognizing whether equations are dependent or independent is a key factor in determining the nature of the solution to a system.

Elimination Method

The elimination method is a systematic approach used to solve systems of linear equations by eliminating one variable at a time. This is done by aligning the equations in such a way that adding or subtracting them cancels out one of the unknowns, ultimately reducing the system to one equation with one variable which can be solved directly.

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 values for these variables that satisfy all equations simultaneously. Such systems often represent intersecting lines in a plane, and their solution corresponds to the point(s) at which these lines intersect.

Consistent vs Inconsistent Systems

A system is termed consistent if there exists at least one set of variable values that satisfies all equations in the system. This includes both systems with a unique solution and those with infinitely many solutions (dependent systems). On the other hand, an inconsistent system has no solution because the equations contradict each other, meaning the graphs of the equations in a plane do not intersect.

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