Solve each system of equations. If the system has no...

Solve each system of equations. If the system has no solution, state that it is inconsistent. Graph the lines of the system.
$$
\left\{\begin{array}{l}
2 x+4 y=\frac{2}{3} \\
3 x-5 y=-10
\end{array}\right.
$$

Key Concepts

Graphical Method

The graphical method involves plotting each equation on a coordinate system and identifying the point where all graphs intersect, which represents the solution to the system. This visual approach not only aids in understanding the solution but also provides insight into the relationship between the equations, such as whether they are parallel (indicating no solution) or coincident (indicating infinitely many solutions).

Intersection Point

In the context of systems of linear equations, the intersection point is the coordinate that satisfies all equations simultaneously. It represents the unique solution when the system is consistent, and the plotted lines intersect at exactly one point.

System of Linear Equations

A system of linear equations consists of two or more equations that share the same set of variables. Solving the system means finding the values of these variables that satisfy all equations simultaneously. This concept is fundamental in algebra and appears in various applications such as physics, economics, and engineering.

Elimination Method

The elimination method is a technique used to solve systems of linear equations by adding or subtracting the equations to eliminate one variable. Once one variable is eliminated, the remaining equation can be solved for the other variable, and its value can then be substituted back to find the first variable. This method is especially useful for systems with two or more equations.

Consistency and Inconsistency

A system of linear equations is consistent if there exists at least one solution (that is, the lines intersect), and inconsistent if there is no solution (for example, when the lines are parallel). Recognizing whether a system is consistent or inconsistent is a key aspect of understanding systems of equations.

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