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
$\left\{\begin{aligned} 2 x-y & =-1 \\ x+\frac{1}{2} y & =\frac{3}{2}\end{aligned}\right.$

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Key Concepts

System Consistency

System consistency refers to whether a system of equations has at least one solution. A consistent system will have one unique solution or infinitely many solutions, while an inconsistent system has no solution at all. Recognizing consistency is crucial when interpreting the results of a system of equations.

Substitution Method

The substitution method requires solving one of the equations for one variable in terms of the others and substituting this expression into the other equation(s). This method is effective for systems where one equation is easily manipulated to isolate a variable, simplifying the process of finding the solution.

Elimination Method

The elimination method involves adding or subtracting the equations in a system to eliminate one variable, making it easier to solve for the other. This technique is particularly useful in systems where coefficients of one variable are opposites or easily made opposites, allowing for a streamlined solution process.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find values for these variables that satisfy all equations simultaneously. This concept is foundational in algebra and is widely applied in areas such as engineering, economics, and physics.

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