Solve each system by elimination or substitution. Label each...

Solve each system by elimination or substitution. Label each system as consistent or inconsistent. Provide reasons for each step. Check the answer.
$$
\begin{aligned}
&\frac{2}{5} P+\frac{5}{2} T=26\\
&\frac{5}{2} P-\frac{2}{5} T=40.71
\end{aligned}
$$

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The objective is to find the values of the variables that satisfy all equations simultaneously. This concept is fundamental in algebra and is widely used in various applications such as physics, economics, and engineering.

Elimination Method

The elimination method involves manipulating the equations in a system (usually through multiplication, addition, or subtraction) to eliminate one or more of the variables. By reducing the system to a simpler form, it becomes easier to solve for the remaining variables. This strategy is particularly useful when the coefficients of one of the variables are opposites or can be made so.

Substitution Method

The substitution method involves solving one of the equations for one variable in terms of the others and then substituting that expression into the remaining equations. This process reduces the number of variables in the subsequent equations, allowing for step-by-step solution of the system. It is especially effective when one equation can be solved easily for one of the variables.

Consistency and Inconsistency in Systems

A system of equations is deemed consistent if there exists at least one set of values that satisfies all equations simultaneously. Conversely, the system is inconsistent if no such set of values exists. Recognizing whether a system is consistent or inconsistent is crucial, as it determines whether a solution exists, whether unique, infinite, or none at all.

Verification of Solutions

Once a solution to a system of equations is obtained, it is essential to substitute the values back into the original equations to verify that they hold true. This process helps to confirm that the steps followed were correct and that the solution is valid. Verification is a key step in ensuring the reliability of the solved system.

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