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.
$$
\left\{\begin{array}{r}
2 x+3 y=6 \\
x-y=\frac{1}{2}
\end{array}\right.
$$

Key Concepts

Consistency and Inconsistency in Systems

A system is consistent if it has at least one solution and inconsistent if it has no solution. In the context of linear equations, inconsistency typically occurs when the lines represented by the equations are parallel and never meet, meaning there is no set of variable values that can satisfy both equations simultaneously.

Graphing Linear Equations

Graphing involves plotting each equation on a coordinate plane and identifying the point(s) at which the graphs intersect. This visual approach shows how the lines (or curves) relate to each other, and the intersection point represents the solution to the system. It also provides an intuitive understanding of unique solutions, infinite solutions, or no solution, depending on whether the lines intersect, coincide, or are parallel.

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. The objective is to find values for these variables that satisfy all equations simultaneously. Systems can have a unique solution, infinitely many solutions, or no solution at all, depending on how the equations relate to each other.

Elimination Method

The elimination method is a technique for solving systems of equations by adding or subtracting the equations to cancel one of the variables. This method simplifies the system into a single equation with one variable, making it easier to solve. It is highly effective when the coefficients of one variable in the equations are opposites or multiples of each other.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting this expression into the other equation. This method reduces a system of equations to a single equation with one unknown, allowing for straightforward solution finding. It is particularly useful when one of the equations is already solved for one variable or can be easily manipulated to do so.

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