What are solutions to a system of linear equations?

Solutions to a system of linear equations are defined as the values of the variables that satisfy all equations in the system simultaneously. This means that for a solution to be considered valid, it must make all equations within the system true at the same time.

For example, in a system of two equations with two variables, finding a solution involves determining the specific values of those variables that result in both equations being true. This concept reflects the graphical interpretation where the solution corresponds to the point(s) where the lines representing the equations intersect on a coordinate plane. If the lines intersect at a point, that point represents the set of values for the variables that solves both equations.

In contrast, other definitions may only address individual equations or fail to capture the idea of multiple equations functioning together, which is not representative of what a solution to the entire system entails. Thus, the focus on simultaneous satisfaction of all equations clearly distinguishes the correct answer in this context.