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The concept of 'systems of equations' refers to a set of two or more equations that have common variables. These systems can be solved using various methods such as substitution, elimination, or graphing. Here, we have a system of linear equations with two variables, s and t. The goal is to find values for s and t that satisfy both equations at the same time. A solution to a system of equations is a set of values for the variables that makes all the equations true. When we graph these equations, the solution is the point(s) where the lines intersect. In algebraic methods, we manipulate the equations to find these values.

Set-builder notation is a mathematical notation used to describe a set by stating the properties that its members must satisfy. In set-builder notation, we specify a set in the form \(\{ x \mid \text{condition} \}\). This means 'the set of all x such that the condition is true.' For example, the set of all real numbers greater than 0 can be written as \(\{ x \mid x > 0 \}\). When a system has an infinite number of solutions, set-builder notation provides a concise way to represent these solutions. The notation involves describing the set of all possible solutions in terms of a parameter, fulfilling the conditions of the original system of equations.

Infinite solutions occur in systems of equations when the equations are dependent, meaning they are essentially the same equation written in different forms. This happens, for example, when one equation is a multiple of another. For a system with infinite solutions, both equations graph as the same line, with every point on the line being a solution. In the context of algebra, after simplifying the system, we end up with a true statement like \(0 = 0\). This indicates that any value satisfying one of the equations will also satisfy the other. In such cases, set-builder notation is used to express the infinite solutions, as there are countless pairs \( (x, y) \) that can fulfill the equations.

No solution occurs in systems of equations when the equations represent parallel lines that never intersect. This happens when the lines have the same slope but different intercepts. Algebraically, when we attempt to solve such systems, we end up with a contradiction, a statement that is always false (e.g., \(0 = 13\)). This contradiction indicates that there is no set of values for the variables that can satisfy both equations simultaneously. Consequently, the system is classified as inconsistent. Understanding this concept helps in identifying systems where no possible solution exists and thus preventing futile attempts to find one.