Verify that the values of the variables listed are solutions...

Verify that the values of the variables listed are solutions of the system of equations.$$
\begin{aligned}
&\left\{\begin{array}{l}
{x-y=3} \\
{\frac{1}{2} x+y=3}
\end{array}\right.\\
&x=4, y=1 ;(4,1)
\end{aligned}
$$

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Key Concepts

Verification of Solutions

Verification involves substituting the proposed solution back into each equation of the system to check if the equations hold true. This process confirms that the candidate solution is accurate and satisfies all conditions imposed by the system, ensuring the reliability of the solution process.

Substitution Method

The substitution method is a technique used to solve systems of equations where one variable is expressed in terms of the other using one of the equations. This expression is then substituted into the other equation, reducing the system to a single equation with one variable. This method is often used for checking or deriving solutions in a systematic way.

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. Understanding such systems is fundamental in algebra and provides a basis for solving more complex problems in mathematics and applied disciplines.

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