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

Solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems 19-30, graph the lines of the system.
$$
\left\{\begin{array}{r}
2 x-3 y=-1 \\
10 x+y=11
\end{array}\right.
$$

Key Concepts

Elimination Method

The elimination method is a systematic approach for solving systems of equations by adding or subtracting equations to cancel one variable. Once one variable is eliminated, the remaining equation can be solved, and its solution substituted back to find the other variable. This method is especially useful for reducing the number of steps required to find the solution.

System of Linear Equations

A system of linear equations consists of two or more equations that include the same set of variables. The goal is to find the values of the variables that satisfy all equations at the same time. These systems form the foundation of algebraic methods and can be solved using various techniques, such as substitution, elimination, or graphical methods.

Solution Consistency

Determining the consistency of a system involves examining whether the equations share a common solution. A system is consistent if there is at least one solution, which can be unique (independent system) or infinite (dependent system); it is inconsistent if no solution exists. In the graphical context, an inconsistent system typically corresponds to parallel lines that never intersect.

Graphing Lines

Graphing is a visual method of solving systems by plotting each equation as a line on the coordinate plane. The point where the lines intersect represents the solution to the system. Graphical representation helps in understanding the nature of the solution, whether the lines intersect at a single point, coincide, or are parallel.

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