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Exponential functions are mathematical expressions that model growth or decay processes. They are of the form \(y = ae^{bx}\), where \(a\) and \(b\) are constants, and \(e\) represents the base of the natural logarithms, approximately equal to 2.71828. The characteristics of exponential functions include:

• Rapid growth or decay: Depending on whether \(b\) is positive or negative, the function will either grow or decay exponentially as \(x\) increases.
• Always positive outputs: When \(a\) and \(b\) are positive, the output will also be positive as \(e^{bx}\) is always greater than zero.

Exponential functions find applications in real-world scenarios such as calculating compound interest, modeling population growth, and understanding radioactive decay. In the context of systems of equations, they provide one of the functions that need to be solved or verified. When dealing with exponential terms like \(e^x\), it is crucial to substitute the value of \(x\) accurately to determine if a particular solution works for the given equation.

Ordered pairs are a fundamental part of understanding solutions to systems of equations. They come in the form \((x, y)\), where \(x\) is the input (or independent variable), and \(y\) is the output (or dependent variable). Test each ordered pair by substituting it into the system to see if the pair is a solution.

• Substitute the values of the ordered pair into all given equations.
• Determine if true statements are achieved for both equations simultaneously.

If an ordered pair satisfies all equations in the system, it is considered a solution. This means both equations yield true statements. Understanding how to test ordered pairs involves plugging values back into the equations, similar to trial and error, to ensure both equations hold true simultaneously.

The solution to a system of equations is an ordered pair (\(x, y\)) that makes all equations in the system true. For systems involving exponential functions like the one in the exercise, verifying solutions requires careful substitution.

• Insert the \(x\) value of the pair into all equations and solve for \(y\).
• See if the resultant \(y\) matches the original \(y\) of the ordered pair.
• If both values in the pair satisfy both equations, then the pair is a solution.

For the given exercise, neither \((-2, 0)\) nor \((-1, 2)\) satisfied both equations of the system, showing they were not solutions. Solving systems typically involves checking each pair systematically until a correct solution is found or confirming that no solution exists. This process helps build strong problem-solving skills and an understanding of linear and nonlinear equations interaction.