Use the substitution or elimination method to solve each system of equations. Identify any inconsistent systems or systems with infinitely many solutions. If a system has infinitely many solutions, write the solution set with y arbitrary.
$\frac{1}{6x}+\frac{1}{3y}=8$
$\frac{1}{4x}+\frac{1}{2y}=12$

Verified step by step guidance

1. First, write the system of equations clearly: \[\frac{1}{6}x + \frac{1}{3}y = 8\] \[\frac{1}{4}x + \frac{1}{2}y = 12\]
2. To eliminate fractions, multiply both sides of each equation by the least common denominator (LCD) of the fractions in that equation. For the first equation, the LCD is 6, so multiply the entire equation by 6: \[6 \times \left( \frac{1}{6}x + \frac{1}{3}y \right) = 6 \times 8\] For the second equation, the LCD is 4, so multiply the entire equation by 4: \[4 \times \left( \frac{1}{4}x + \frac{1}{2}y \right) = 4 \times 12\]
3. Simplify both equations after multiplication to get rid of the fractions. This will give you two linear equations in standard form:
4. Use either the substitution or elimination method to solve the system. For elimination, align the equations and multiply one or both equations if necessary to get coefficients of either \(x\) or \(y\) to be opposites, then add or subtract the equations to eliminate one variable.
5. After eliminating one variable, solve for the remaining variable. Substitute this value back into one of the original equations (or the simplified ones) to find the other variable. Check if the system is consistent (one unique solution), inconsistent (no solution), or has infinitely many solutions (dependent equations). If infinitely many solutions occur, express the solution set with \(y\) as an arbitrary parameter.

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Solutions can be unique, infinite, or nonexistent depending on the relationships between the equations.

Substitution and Elimination Methods

Substitution involves solving one equation for a variable and substituting it into the other equation, while elimination involves adding or subtracting equations to eliminate a variable. Both methods simplify the system to find the values of variables efficiently.

Types of Solutions: Consistent, Inconsistent, and Dependent Systems

A consistent system has at least one solution; it can be unique or infinite. An inconsistent system has no solution, often due to parallel lines. Dependent systems have infinitely many solutions, where equations represent the same line, and solutions are expressed with a parameter.