Question

If \(w=g(x, y)=7-x y\) find the value of the dependent variable \(w\) when \(x=-3\) and \(y=-9\)

Short answer

Answer: -20

Step-by-step solution

Identify the function g(x, y) and the given values of x and y

We are given the function g(x, y) = 7 - xy with x = -3 and y = -9.

Plug in the given values of x and y into the function

Now we substitute the given values of x and y into the function g(x, y): g(-3, -9) = 7 - (-3)(-9)

Solve for w

Now we just need to perform the arithmetic to find the value of w: g(-3, -9) = 7 - 27 = -20 Therefore, when x = -3 and y = -9, the value of the dependent variable w is -20.

Key concepts

Understanding the Dependent Variable

In multivariable calculus, a dependent variable is the result of a function that relies on one or more other variables. For example, in the exercise provided, the dependent variable is represented by \( w \), which is defined by the function \( g(x, y) = 7 - xy \). Here, \( w \) depends on both \( x \) and \( y \).

The role of dependent variables in mathematics is critical. They offer a way to express how changes in input variables affect the outcome. In real-life situations, dependent variables could represent anything from cost based on production parameters to temperature changes in response to environmental factors.

In this specific exercise, identifying \( w \) as the dependent variable helps to clarify its reliance on the values of \( x \) and \( y \), thus adding clarity to the problem-solving process of evaluating the function.

Function Evaluation Basics

Function evaluation is a fundamental concept in mathematics, including multivariable calculus. It involves calculating the value of a function given specific inputs. In our exercise, the function \( g(x, y) = 7 - xy \) is evaluated by substituting the given values of \( x = -3 \) and \( y = -9 \).

To perform a function evaluation:

• First, identify the given function and its algebraic expression.
• Substitute the provided values for each variable in the function.
• Calculate the expression to find the result.

In this example, substituting yields \( g(-3, -9) = 7 - (-3)(-9) \). Calculating the expression leads us to \( 7 - 27 = -20 \). Thus, function evaluation simplifies a complex equation and provides the essential outcome associated with specific input values.

Substitution Method

The substitution method is a straightforward yet powerful technique used to solve equations by replacing variables with given values. This technique simplifies the process of understanding and solving complex problems in multivariable calculus.

Here’s how substitution works in our example:

• Start with the original equation or function, which is \( g(x, y) = 7 - xy \).
• Substitute the specified values for each variable. In the exercise, we apply \( x = -3 \) and \( y = -9 \).
• Simplify the equation to reach a solution. The substitution transforms the original function into \( g(-3,-9) = 7 - (-3)(-9) \).

Through substitution, the initial problem becomes a simple arithmetic problem, making it easier to solve. This method is valuable in numerous situations where evaluating a function's outcome with specific parameters is necessary.