What constitutes a solution of a system of equations?

A solution of a system of equations is defined as an ordered pair that satisfies all equations in the system simultaneously. When a pair of values, typically denoted as (x, y), is substituted into each equation, it should make all equations true at the same time. This means that each equation holds valid when these values are applied, confirming that they are indeed solutions to the entire system, not just individual equations.

In contrast, a variable that makes only one equation true does not account for the other equations in the system, so it cannot be considered a solution to the system as a whole. Solutions limited to only positive integer pairs exclude many valid solutions that may include negative numbers or fractions. A random combination of numbers does not ensure any relation to the equations in the system, making it unlikely to satisfy the conditions required for solutions. Therefore, the definition of a solution as an ordered pair that satisfies all equations reflects the necessary criteria accurately.