Question

Given \(z=f(x, y)=7 x+2 y\) find the output when \(x=8\) and \(y=2\).

Short answer

Answer: The value of the function when \(x = 8\) and \(y = 2\) is \(z = 60\).

Step-by-step solution

Identify the function

We are given the function \(z = f(x, y) = 7x + 2y\) and need to find the value of the function when \(x = 8\) and \(y = 2\).

Substitute the values of x and y in the function

Replace the value of \(x\) with \(8\) and the value of \(y\) with \(2\) in the function: \(z = 7(8) + 2(2)\).

Compute the value of the function

Now, calculate the value of the function: \(z = 7(8) + 2(2) = 56 + 4 = 60\).

Write the final answer

The value of the function \(z = f(x, y) = 7x + 2y\) when \(x = 8\) and \(y = 2\) is \(z = 60\).

Key concepts

Function Substitution

Understanding function substitution is crucial when dealing with multivariable functions. It's a method that allows us to insert specific values in place of variables within the function's formula and is particularly useful for evaluating the function at given points. The process of function substitution involves replacing the variables in the function's expression with the given numbers or expressions.
For example, suppose we have a function defined as \( z = f(x, y) = 7x + 2y \). To evaluate this function for \( x = 8 \) and \( y = 2 \), we perform the substitution by plugging these values into the function:
\( z = 7(8) + 2(2) \).
This is the essence of function substitution: a straightforward replacement technique that brings us one step closer to determining the function's output for the given inputs.

Calculating Function Value

After substituting the values into the function, the next step is calculating the function value. This involves basic arithmetic operations such as addition, multiplication, or more complex processes depending on the function's form. For our linear function example, \( z = f(x, y) = 7x + 2y \), the calculation after substitution is simple.
With \( x = 8 \) and \( y = 2 \), we multiply and add as follows:
\( z = 7(8) + 2(2) = 56 + 4 \).
By performing these arithmetic operations, we find that \( z = 60 \). This resulting number is the specific output of the function corresponding to the substituted inputs. Computing the function’s value accurately is crucial to understanding the relationship between the inputs and the output in various applications, from basic algebra to complex sciences.

Linear Functions

Linear functions are fundamental in mathematics and represent a relationship between variables where the graph is a straight line. These functions have a constant rate of change and are described by an equation of the form \( y = mx + b \), where 'm' is the slope and 'b' is the y-intercept.
In multivariable calculus, the concept of linear functions extends to functions of several variables like \( z = f(x, y) = 7x + 2y \), which is the function we're analyzing. In a linear function of two variables, 'x' and 'y' are inputs, and 'z' is the output. The coefficients of 'x' and 'y' in this context are the slopes along the respective axes, indicating how the function's value changes as the inputs vary.
By exploring linear functions, students learn to predict and understand linear relationships within real-world contexts. The process of evaluating these functions, as performed in our example, is essential for students to master as they provide a strong foundation for understanding more complex relationships in higher-dimensional spaces.