Problem 72 - Algebra A Combined Function

Question

Evaluate each expression for the given values. See Section 1.8 \(r \cdot t ; \quad r=15\) and \(t=2\)

Step-by-Step Solution

Answer

The value of the expression is 30.

Step 1: Identify the Expression and Variables

The expression given is \(r \cdot t\), where \(r\) and \(t\) are variables. We are given specific values for these variables: \(r = 15\) and \(t = 2\).

Step 2: Substitute the Values Into the Expression

Substitute the given values of \(r\) and \(t\) into the expression. This means replacing \(r\) with 15 and \(t\) with 2 in the expression \(r \cdot t\). Thus, our expression becomes \(15 \cdot 2\).

Step 3: Perform the Multiplication

Multiply the numbers obtained after substitution: \(15 \cdot 2 = 30\). This is the evaluation of the expression with the substituted values.

Key Concepts

Substitution Method

The substitution method is a fundamental technique used to evaluate algebraic expressions. It involves replacing variables with given numerical values. Let's take a closer look:

Imagine you have an expression like \(r \cdot t\). Here, \(r\) and \(t\) are variables, standing in for actual numbers. In an exercise, you might be asked to evaluate this expression for specific values, such as \(r = 15\) and \(t = 2\).

To use the substitution method, follow these steps:

• Identify the expression and the variables involved.
• Take the values given for each variable. In this case, \(r = 15\) and \(t = 2\).
• Substitute, or replace, each variable in the expression with its corresponding value. This will transform your expression into an arithmetic problem, such as \(15 \cdot 2\).

By the end of this process, you've turned an abstract expression into a concrete calculation that you can easily solve.

Multiplication

Multiplication is a basic arithmetic operation that involves adding a number, called the multiplicand, to itself a certain number of times determined by another number, called the multiplier. When you multiply two numbers, you'll get a product.

For example, in the expression \(r \cdot t\), once substitution is complete, you get \(15 \cdot 2\).

Here's how multiplication works in this context:

• Interpret "\(\cdot\)" as the operation of multiplication.
• In the problem \(15 \cdot 2\), you calculate by adding fifteen twice: \(15 + 15 = 30\).
• While simple, multiplying correctly is crucial in evaluating expressions accurately.

This concept extends to more complex operations and helps lay the groundwork for understanding more advanced mathematical algorithms and operations. Remember, multiplication forms the basis of many arithmetic procedures and mathematical reasoning.

Variables in Algebra

Variables are symbols used to represent unknown or unspecified numbers in mathematical expressions and equations. They allow for flexibility and generality in equations.

Here's why variables are vital in algebra:

• Variables can stand for numbers that might change or be unknown at first but are later specified, like \(r\) and \(t\).
• They help in creating formulas and equations. For example, \(r \cdot t\) can represent countless calculations where only \(r\) and \(t\) differ.
• Variables are the building blocks in algebra that help in forming expressions, which can then be manipulated to find solutions or understand relationships between numbers.

By becoming comfortable with seeing and using variables, students improve their ability to decipher and solve mathematical problems across different contexts and applications.