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Substitution in functions is a powerful tool in mathematics that allows us to simplify and solve complex equations. When dealing with a multivariable linear function, like the one given, each variable must be replaced with its specified value. In our function, \(y = 78.1 + 0.83 x_{1} - 0.09 x_{2} + 1.19 x_{3}\), substitution starts by taking each of the specified values: \(x_1 = 77\), \(x_2 = 0\), and \(x_3 = 68\), and inserting them into their respective places in the function.This results in the modified equation: \[ y = 78.1 + 0.83(77) - 0.09(0) + 1.19(68) \].Substitution helps transform an abstract equation into a concrete solution. Without substitution, understanding the relation and solving the function for specific variables wouldn't be possible. By methodically following this process, complex functions become manageable, and their outcomes understandable.

Solving mathematical problems, particularly those involving multiple variables, requires a clear, orderly approach. The step-by-step method is an organized way to tackle these problems by breaking them down into smaller, more manageable parts.1. **Identify Given Values**: The first and most crucial step is recognizing your specific variables and values. In our exercise, these are \(x_1 = 77\), \(x_2 = 0\), and \(x_3 = 68\).2. **Substitute Values**: Once identified, substitute these values into the equation, transforming it from a general to a specific instance.3. **Calculate Each Term**: Solve each component of the equation separately. This includes multiplying coefficients by their corresponding variables: \(0.83 \times 77\), \(-0.09 \times 0\), and \(1.19 \times 68\).4. **Sum All Parts**: Finally, add all parts, including any constants, to find your solution.Step-by-step problem solving ensures clarity and accuracy, minimizing errors while advancing toward a precise conclusion.

Evaluating mathematical expressions involves computing the value of an expression once variables have been replaced by specific values. This process is essential to determine the numeric result of an otherwise abstract mathematical equation.For our function, after substitution, we handled each calculation individually:

• Calculate \(0.83 \times 77 = 63.91\)
• Compute \(-0.09 \times 0 = 0\)
• Find \(1.19 \times 68 = 80.92\)

The next step is combining these results with the constant \(78.1\), ensuring precision in each arithmetic operation.Finally, all these results are summed: \(78.1 + 63.91 + 0 + 80.92 = 222.93\). This procedure not only confirms the value of the expression but also enhances comprehension of how each variable affects the outcome. Evaluating such expressions allows for clearer insights and aids in making well-informed conclusions about the function's behavior.