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Linear functions are remarkable for their straightforward form and predictable behavior. A linear function is any function that can be written in the format \(f(x) = ax + b\). Here, \(a\) and \(b\) are constants. The graph of a linear function is always a straight line. The slope of this line is determined by the constant \(a\), which describes the steepness and direction of the line:

• If \(a\) is positive, the line rises as it moves from left to right.
• If \(a\) is negative, the line falls as it moves from left to right.

The constant \(b\) represents the y-intercept, the point where the line crosses the y-axis. This means that when \(x = 0\), \(f(x) = b\). Linear functions are commonly used in a variety of real-world situations, such as modeling profit-loss scenarios and predicting growth rates. Understanding them is all about seeing how each part of their equation contributes to the overall shape and position of the line.

The substitution method is a fundamental technique used in mathematics to find the value of a function at a specific input. This technique is straightforward yet powerful. It involves replacing the variable in the function with the given input value. For example, if you have a function \(g(x) = 3x + 5\) and you want to find \(g(-5)\), simply replace \(x\) with \(-5\). It's like following a recipe, where you replace general ingredients with specific amounts. In this exercise, substituting \(-5\) into the function results in:

• \(g(-5) = 3(-5) + 5\)

The substitution method simplifies calculations and allows us to evaluate functions with ease. This technique is especially useful in algebra and calculus, providing a systematic approach to finding solutions.

Solving equations is a core skill in mathematics, aiding in finding unknown values within an equation. In this context, after we substitute a value into a function, the next step is solving the expression. Consider the function \(g(x) = 3x + 5\). After substituting \(-5\) for \(x\), as described in the substitution method, we have the expression \(3(-5) + 5\).
To solve it:

• First, multiply \(3\) by \(-5\) to get \(-15\).
• Then, add \(5\) to \(-15\), resulting in \(-10\).

The solution \(-10\) represents the output of the function when \(x = -5\). Solving equations like these helps in simplifying complex problems by breaking them down into manageable calculations. It showcases the importance of understanding mathematical operations and order in obtaining a solution.