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.
$$
\left\{\begin{array}{l}
\frac{1}{x}+\frac{1}{y}=8 \\
\frac{3}{x}-\frac{5}{y}=0
\end{array}\right.
$$

Key Concepts

System Consistency

Understanding whether a system is consistent (has at least one solution) or inconsistent (has no solution) is crucial. A system may also have infinitely many solutions or a unique solution, and identifying this characteristic is a key part of solving the system effectively.

System of Equations

A set of equations with multiple variables that need to be solved simultaneously, meaning that the solution must satisfy every equation in the system. Solving such systems requires methods like substitution or elimination to find the values of the variables that make all the equations true at the same time.

Domain Restrictions

When working with equations involving terms like 1/x or 1/y, it is essential to note that the variables cannot equal zero because division by zero is undefined. Recognizing these restrictions ensures that only valid solutions are considered.

Reciprocal Transformation

This concept involves redefining variables that appear as reciprocals (for example, setting a = 1/x and b = 1/y) to transform the given equations into a simpler linear form. This method helps in utilizing standard techniques for solving linear systems.

Substitution and Elimination Methods

These are common techniques used to solve systems of equations. The substitution method involves expressing one variable in terms of another from one equation and substituting it into the other equations. The elimination method involves adding or subtracting equations to cancel out one variable, thereby simplifying the system to find the remaining variables.

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