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.
$$
\left\{\begin{array}{l}
\frac{4}{x}-\frac{3}{y}=0 \\
\frac{6}{x}+\frac{3}{2 y}=2
\end{array}\right.
$$

Key Concepts

System of Equations

A system of equations consists of two or more equations with the same set of variables, and the goal is to find values for these variables that satisfy all of the equations simultaneously. It is fundamental in mathematics to analyze whether the system has a unique solution, infinitely many solutions, or no solution.

Rational Equations

Rational equations involve fractions with polynomials in the numerator and/or denominator. When dealing with such equations, it is important to identify and consider the restrictions on the variables, particularly values that would make any denominator zero, to ensure that all solutions are valid.

Variable Substitution and Transformation

In problems where variables appear as their reciprocals, a common tactic is to substitute new variables (for example, letting u = 1/x and v = 1/y). This transformation converts the original rational system into a linear system, which is often easier to manipulate and solve using standard techniques.

Graphical Interpretation of Linear Systems

Graphing the lines represented by the system of equations provides a visual method of finding the solution by locating the intersection point(s) of the lines. This visual approach is not only useful for identifying unique solutions but also for detecting cases where the system is inconsistent, such as when the lines are parallel and do not intersect.

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