In Problems 17-54, solve each system of equations. If the...

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{array}{l}\frac{1}{3} x-\frac{3}{2} y=-5 \\ \frac{3}{4} x+\frac{1}{3} y=11\end{array}\right.$

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The goal is to find the values of the variables that satisfy all equations simultaneously. Such systems can have a unique solution, infinitely many solutions, or no solution at all, depending on the relationship between the equations.

Elimination Method

The elimination method is a technique used to solve systems of linear equations by combining equations to cancel out one variable, thereby reducing the system to a single equation with one unknown. This approach involves aligning coefficients and adding or subtracting the equations, which simplifies the solving process.

Consistent and Inconsistent Systems

A consistent system of equations has at least one solution, meaning there is a set of values that satisfies all the equations. In contrast, an inconsistent system has no solution because the equations contradict each other. Recognizing these cases is important to determine whether a system can be solved or labeled as inconsistent.

Clearing Fractions

When equations include fractional coefficients, clearing the fractions by multiplying each term by the least common denominator can simplify the problem. This technique converts fractional equations into equivalent equations with integer coefficients, making it easier to apply methods such as elimination or substitution.

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