Problem 7 - College Algebra with Modeling and Visualization

Question

Evaluate the expression for the given \(f(x, y)\). $$ f\left(\frac{1}{2},-\frac{7}{4}\right) \text { if } f(x, y)=\frac{2 x}{y+3} $$

Step-by-Step Solution

Answer

The result is \( \frac{4}{5} \).

Step 1: Understand the Expression

We need to evaluate the expression given by the function \( f(x, y) = \frac{2x}{y+3} \). This involves substituting specific values for \( x \) and \( y \) into the function.

Step 2: Identify Given Values

The exercise provides specific values for \( x \) and \( y \). We are given \( x = \frac{1}{2} \) and \( y = -\frac{7}{4} \). These values must be substituted into the function.

Step 3: Substitute Values into the Function

Substitute \( x = \frac{1}{2} \) and \( y = -\frac{7}{4} \) into the function: \[ f\left( \frac{1}{2}, -\frac{7}{4} \right) = \frac{2 \left(\frac{1}{2}\right)}{-\frac{7}{4} + 3} \]

Step 4: Simplify the Numerator

Calculate the numerator: \[ 2 \left(\frac{1}{2}\right) = 1 \]

Step 5: Simplify the Denominator

Calculate the denominator: \[ -\frac{7}{4} + 3 = -\frac{7}{4} + \frac{12}{4} = \frac{5}{4} \]

Step 6: Evaluate the Expression

Replace the numerator and denominator and simplify: \[ f\left(\frac{1}{2}, -\frac{7}{4}\right) = \frac{1}{\frac{5}{4}} = 1 \cdot \frac{4}{5} = \frac{4}{5} \]

Step 7: Result

The evaluated result of the expression is \( \frac{4}{5} \).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition and multiplication). These are used to represent mathematical situations. In this exercise, the function \( f(x, y) = \frac{2x}{y+3} \) is an algebraic expression involving two variables, \( x \) and \( y \).
Understanding algebraic expressions is crucial. They allow us to form equations and functions that can model real-world scenarios. In functions such as this one, any change in \( x \) or \( y \) will affect the output, showing the relationship between these variables and their combined effects. By recognizing that \( 2x \) and \( y + 3 \) are parts of our expression, we know these components will dictate how we find the function's output.

Substitution Method

The substitution method is a key technique in algebra to evaluate expressions like \( f(x, y) = \frac{2x}{y+3} \). When given specific values for variables, we replace the variables in the expression with these values to find a specific result. In this exercise, we've been given \( x = \frac{1}{2} \) and \( y = -\frac{7}{4} \).
Substituting these values involves:

• Replacing \( x \) with \( \frac{1}{2} \) anywhere it appears in the expression.
• Replacing \( y \) with \( -\frac{7}{4} \).

After the substitution, the expression becomes \( f\left( \frac{1}{2}, -\frac{7}{4} \right) = \frac{2(\frac{1}{2})}{-\frac{7}{4} + 3} \). This transformation makes it possible to simplify and evaluate the expression to find an exact value.

Simplification Techniques

Simplification techniques in algebra help us reduce an expression to its simplest form. After substituting values into our function \( f(x, y) = \frac{2x}{y+3} \), we must simplify both the numerator and the denominator.
For the numerator:\

• We multiply \( 2 \times \frac{1}{2} \), which simplifies to \( 1 \).

For the denominator:

• First, calculate \( -\frac{7}{4} + 3 \) by recognizing \( 3 \) as \( \frac{12}{4} \).
• Add these fractions: \(-\frac{7}{4} + \frac{12}{4} = \frac{5}{4} \).

By dividing the simplified numerator by the denominator, \( \frac{1}{\frac{5}{4}} \), we use flipping and multiplying to get \( 1 \times \frac{4}{5} \), which simplifies to \( \frac{4}{5} \). These step-by-step simplifications ensure an accurate result through clear operations.