Problem 69 - Elementary and Intermediate Algebra

Question

Evaluate. $$ 45 \div 3 \cdot a, \text { for } a=-1 $$

Step-by-Step Solution

Answer

-15

Step 1 - Substitute the variable

Replace the variable \( a \) with its given value, which is \( -1 \). Thus, rewrite the expression as: \[ 45 \div 3 \cdot (-1) \]

Step 2 - Perform the division

First, perform the division operation: \( 45 \div 3 \). This simplifies to 15. Now the expression is: \[ 15 \cdot (-1) \]

Step 3 - Perform the multiplication

Next, multiply 15 by the substituted value \( -1 \): \( 15 \cdot (-1) = -15 \)

Key Concepts

Substitution

Substitution is the process of replacing a variable in an expression with a given value. This is often the first step in evaluating an algebraic expression. For example, if you have the expression \( 45 \div 3 \cdot a \) and are given that \( a = -1 \), you substitute \( -1 \) in place of \( a \). The new expression becomes:
\[ 45 \div 3 \cdot (-1) \]
Substitution is important because it allows you to simplify and solve the expression step-by-step. Always ensure the value you substitute is correct, as a wrong substitution can lead to incorrect results.

Order of Operations

The order of operations is a set of rules that determines the correct sequence to evaluate a mathematical expression. The rules can be remembered using the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

For our example, \( 45 \div 3 \cdot (-1) \), follow these steps:

• First, do the division: \( 45 \div 3 = 15 \)
• Next, perform the multiplication: \( 15 \cdot (-1) = -15 \)

By following the order of operations, you ensure that expressions are evaluated accurately. Forgetting this order can lead to errors in your final result.

Evaluating Expressions

Evaluating an expression means finding its value by performing the given operations. This involves substitution and following the order of operations. Here's the step-by-step process:
1. Substitute the given value for the variable. In the expression \( 45 \div 3 \cdot a \) with \( a = -1 \), substitute \( -1 \) to get:
\[ 45 \div 3 \cdot (-1) \]
2. Follow the order of operations—perform division first:
\( 45 \div 3 = 15 \)
3. Then, multiply the result by the substituted value:
\( 15 \cdot (-1) = -15 \)
Always check your steps to ensure accuracy. Evaluating expressions correctly helps you solve many algebraic problems.