Problem 36 - Intermediate Algebra

Question

Evaluate the algebraic expressions for the given values of the variables. $$ 5 x-9 y, \quad x=-2 \text { and } y=5 $$

Step-by-Step Solution

Answer

The evaluated expression is -55.

Step 1: Evaluate 5x

To find the value of the expression, substitute the given value of \( x = -2 \) into the expression. Calculate \( 5x \) as follows: \[5(-2) = -10.\]

Step 2: Evaluate -9y

Now substitute the given value of \( y = 5 \) into the expression and calculate \( -9y \):\[-9(5) = -45.\]

Step 3: Combine Results

Now combine the results from the previous steps to evaluate the entire expression:\[5x - 9y = -10 - 45.\]Calculate the final result.\[-10 - 45 = -55.\]

Key Concepts

Substitution

Substitution is a fundamental concept in algebra that involves replacing variables in an expression with their given numerical values. This process allows us to simplify expressions and ultimately find their values. In the given exercise, we were provided with the algebraic expression \( 5x - 9y \) and specific values for the variables: \( x = -2 \) and \( y = 5 \).

To substitute, you need to identify where each variable is located within the expression. Once each variable is identified, replace it with the provided value. For instance:

• For \( x \), replace it with \(-2\), resulting in \( 5(-2) \).
• For \( y \), replace it with \( 5 \), resulting in \(-9(5)\).

This substitution sets the stage for the evaluation of the expression. The next steps are focused on performing arithmetic calculations after these substitutions to find a simplified form or a numerical result.

Evaluation

Once substitution is complete, the next step in simplifying and solving an algebraic expression is evaluation. Evaluation involves performing the arithmetic operations dictated by the expression with the substituted values.

In the given example, after substituting \( x = -2 \) and \( y = 5 \), the expression became \( 5(-2) - 9(5) \).

Let's break down the evaluation into clear steps:

• First, calculate \( 5(-2) \), which equals \(-10\).
• Next, calculate \(-9(5)\), which equals \(-45\).

The evaluation step combines these calculations. To finalize, sum up the results: \(-10 - 45\).

This arithmetic operation leads to the final result of \(-55\). Evaluation gives us the numerical value of an entire expression when specific values are substituted for the variables.

Variables

The concept of variables is central to algebra and forms the building blocks of algebraic expressions. Variables are symbols, often letters like \( x \) or \( y \), that represent numbers whose values are not specified within the expression. This allows expressions to remain general and applicable to a variety of situations.

In the exercise, we worked with the variables \( x \) and \( y \). These variables were placeholders for numbers, which were later revealed to be \(-2\) and \( 5 \), respectively. Understanding variables involves recognizing that:

• Variables can take on any value from a set range, facilitating the exploration of multiple scenarios.
• They make expressions flexible, adaptable, and capable of representing general mathematical relationships.
• Ultimately, when specific values are assigned to variables, it enables us to calculate and evaluate expressions numerically.

By recognizing the role of variables, you better understand how algebra allows for the modeling and solving of real-world problems through abstract symbolic representation.