Determine the Number of Solutions of a Linear System Without...

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.
$$
\left\{\begin{array}{l}
4 x+2 y=10 \\
4 x-2 y=-6
\end{array}\right.
$$

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Key Concepts

Types of Solutions (Unique, None, Infinitely Many)

Linear systems can have a unique solution, meaning a single point of intersection, infinitely many solutions when the equations represent the same line (coincident), or no solution if the lines are parallel and distinct. This classification is fundamental in understanding the behavior of the system.

Consistent vs. Inconsistent Systems

A consistent system is one that has at least one solution, whether unique or infinitely many, while an inconsistent system has no solutions. Determining consistency helps in understanding whether the equations agree on a common point of intersection.

Linear Systems of Equations

A linear system of equations consists of two or more linear equations involving the same set of variables. It represents geometrically as lines (in two dimensions) or planes (in higher dimensions) and is analyzed to determine the points at which these geometric objects intersect.

Elimination Method

The elimination method is an algebraic technique used to solve linear systems by adding or subtracting equations to eliminate one of the variables. This approach simplifies the system to a single equation with one variable, facilitating the determination of the solution's nature.

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