In Exercises 1-10, determine whether the given ordered pair...

In Exercises $1-10$, determine whether the given ordered pair is a solution of the system.
$$\begin{aligned}
&(1400,450)\\
&\left\{\begin{aligned}
x-2 y &=500 \\
0.03 x+0.02 y &=51
\end{aligned}\right.
\end{aligned}$$

Key Concepts

System of Linear Equations

A system of linear equations is a set of equations with multiple variables where each equation represents a line in a coordinate plane. The goal is to find the variable values that satisfy all equations simultaneously. Understanding such systems is essential in algebra because they often model real-life problems where multiple conditions must be met at the same time.

Ordered Pairs

An ordered pair, typically represented as (x, y), is a fundamental element in coordinate geometry. It specifies a unique point in the plane with one coordinate along the horizontal axis (x) and another along the vertical axis (y). Recognizing ordered pairs is crucial when determining the solutions to equations and systems thereof, as each solution corresponds to a point that satisfies the conditions set by the equations.

Substitution Method

The substitution method is a technique used to determine if an ordered pair is a solution to a system of equations by directly replacing the variables in the equations with specific numeric values. This method is foundational in solving systems because it provides a straightforward way to test if the equations hold true for given candidates, thereby confirming their validity as solutions.

Solution Verification

Solution verification involves substituting the given values of the variables back into each equation in the system to check if they satisfy all the conditions. This process confirms the accuracy of a proposed solution. In algebra, verifying solutions by substitution is a standard method to ensure that the computed ordered pair actually lies on all corresponding lines in the system.

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