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Substitution in algebra is a method used to replace variables with their corresponding numerical values in mathematical expressions or equations. It helps simplify the problem-solving process. When evaluating a function like a multivariable linear function, knowing the substitution technique is crucial.
For example, in the function given: \[ y = 14.7 + 1.4x_1 - 7.8x_2 \] we need to substitute the given values of the variables: \( x_1 = 0 \) and \( x_2 = 21 \). This involves replacing each occurrence of the variables \( x_1 \) and \( x_2 \) in the equation with 0 and 21, respectively.

• The expression becomes: \[ y = 14.7 + 1.4(0) - 7.8(21) \]
• Notice how substitution simplifies the equation into a more manageable form for calculation.

Substitution is a foundational skill in algebra, serving as a tool for evaluating expressions and solving equations efficiently. Understanding substitution makes the transition to more complex algebraic concepts much smoother.

A linear function is a type of mathematical expression where each variable is multiplied by a constant, and these products are summed up with a constant term to form the function.
Given the function: \[ y = 14.7 + 1.4x_1 - 7.8x_2 \] you will notice that each term has a specific role:

• \( 14.7 \) is the constant term, which acts as the initial value of \( y \) when \( x_1 \) and \( x_2 \) are zero.
• \( 1.4x_1 \) and \(-7.8x_2\) are the variable terms, where 1.4 and -7.8 are the coefficients of \( x_1 \) and \( x_2 \), respectively.

To perform linear function calculation, substitute given values for the variables into the equation, simplify each term, and then sum all terms to get the result for \( y \). This step-by-step approach ensures accuracy and clarity, making it easier to handle more complex functions as you advance in math.

Explanatory variables, often referred to as independent variables, play a crucial role in functions and modeling. They are the inputs that we manipulate or control to see how they affect the output or dependent variable, in this case, \( y \).
In the context of the given function: \[ y = 14.7 + 1.4x_1 - 7.8x_2 \] \( x_1 \) and \( x_2 \) serve as explanatory variables. Their values are vital in determining the outcome of the function.

• The value of \( x_1 \) doesn't affect the result when \( x_1 = 0 \) because multiplying anything by zero results in zero.
• Conversely, \( x_2 = 21 \) significantly impacts the result; as \( -7.8 \times 21 = -163.8 \), it heavily influences the final outcome for \( y \).

Understanding the role of explanatory variables helps predict and explain changes within mathematical models and real-life phenomena. Recognizing how these variables interact is fundamental in analyzing functions and data-driven scenarios.