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.
$$
\left\{\begin{array}{l}
\frac{1}{2} x+\frac{1}{3} y=3 \\
\frac{1}{4} x-\frac{2}{3} y=-1
\end{array}\right.
$$

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple equations with multiple variables that are all linear. These systems are solved to find values of the variables that simultaneously satisfy all the equations. Understanding these systems is fundamental in algebra and is widely applied in various fields including engineering, economics, and the sciences.

Elimination Method

The elimination method is a technique used to solve systems of linear equations by combining equations to cancel out one of the variables. This method involves aligning the equations and often multiplying one or more equations by constants to create coefficients that allow one variable to be eliminated, simplifying the system to a single equation with one unknown.

Clearing Fractions

When dealing with systems of linear equations that include fractional coefficients, it is often useful to clear the fractions by multiplying each term by the least common denominator. This process transforms the equations into equivalent forms with integer coefficients, making the arithmetic simpler and reducing potential calculation errors.

Inconsistent System

An inconsistent system of equations is one that has no solution, meaning there is no set of variable values that simultaneously satisfies all the equations. This typically occurs when the equations represent parallel lines (in the case of two variables) that never intersect, highlighting a lack of common solution among the equations.

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