Problem 6 - College Algebra with Modeling and Visualization

Question

Evaluate the expression for the given \(f(x, y)\). $$ f(5,-2) \text { if } f(x, y)=6 y-\frac{1}{2} x $$

Step-by-Step Solution

Answer

The value of \( f(5, -2) \) is \(-14.5\).

Step 1: Understand the Expression

The function provided is \( f(x, y) = 6y - \frac{1}{2}x \). This function is defined with two variables \( x \) and \( y \). We are tasked with evaluating this function for specific values of \( x \) and \( y \).

Step 2: Substitute Values

Substitute \( x = 5 \) and \( y = -2 \) into the expression: \[ f(5, -2) = 6(-2) - \frac{1}{2}(5) \].

Step 3: Calculate Each Term

First, calculate \( 6(-2) \) resulting in \(-12\). Then calculate \( \frac{1}{2}(5) \) resulting in \( 2.5 \).

Step 4: Evaluate the Expression

Now subtract the results from Step 3: \[-12 - 2.5 = -14.5 \]. Thus, \( f(5, -2) = -14.5 \).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They do not have an equals sign, which differentiates them from equations. In the context of our exercise, the expression is part of a function: - The function here is expressed as \( f(x, y) = 6y - \frac{1}{2}x \). - It is a combination of constants (6 and \(\frac{1}{2}\)), variables \(x\) and \(y\), and arithmetic operations such as multiplication and subtraction.Algebraic expressions are foundational to understanding algebra as they allow us to generalize mathematical reasoning and apply it to different situations by changing the variables' values.

Variables Substitution

Variable substitution is a key step in evaluating algebraic expressions within a function. It involves replacing variables with their given values.In our exercise:

• We start with the variables \(x\) and \(y\) which are parts of the expression \(6y - \frac{1}{2}x\).
• The task is to substitute or replace these variables with specified numerical values: \(x = 5\) and \(y = -2\).

After substitution, a numeric expression is formed: \[ f(5, -2) = 6(-2) - \frac{1}{2}(5) \].This step transforms the problem from a general form into one that can be solved with basic arithmetic, allowing us to simplify and find the exact value of the function for given inputs.

Basic Arithmetic Operations

Basic arithmetic operations are used to simplify and solve expressions after variable substitution. These operations include addition, subtraction, multiplication, and division. In this exercise, we specifically engage with:- **Multiplication:** Calculate \(6(-2)\), which equals \(-12\). Here, 6 is multiplied by \(-2\) to apply the operation of scaling a number.- **Division (in form of multiplication by fraction):** Calculate \(\frac{1}{2}(5)\), which equals \(2.5\). This involves multiplying 5 by \(\frac{1}{2}\) (essentially dividing by 2).Finally, the subtraction operation is applied: \[-12 - 2.5 = -14.5\].Understanding these basic operations is crucial, as it enables the transformation of algebraic expressions into a numerical result after substitution. This process helps not only in solving homework problems but in sharpening arithmetic skills useful in everyday math.