Problem 4 - Prealgebra

Question

Evaluate each expression if \(a=5, b=12,\) and \(c=4\) $$18-3 c$$

Step-by-Step Solution

Answer

The expression evaluates to 6.

Step 1: Substitute the value of c

First, substitute the given value of \(c\) into the expression. The original expression is \(18 - 3c\). Replace \(c\) with \(4\), which gives us \(18 - 3 \times 4\).

Step 2: Multiply

Now, calculate \(3 \times 4\). This multiplication gives us \(12\). So, the expression becomes \(18 - 12\).

Step 3: Perform Subtraction

Finally, subtract \(12\) from \(18\). This results in \(18 - 12 = 6\). Therefore, the value of the expression is \(6\).

Key Concepts

Substitution Method

The substitution method is a fundamental concept in algebra that simplifies expressions by replacing variables with known values. This technique is particularly useful when solving equations or evaluating expressions.

In the given exercise, the expression is initially written as \(18 - 3c\), where \(c\) is a variable. By substituting \(c\) with its known value, \(4\), we effectively remove the variable from the expression and convert it into a simpler arithmetic problem. This step transforms the expression into \(18 - 3 \times 4\).

• Identify the variable in the expression and its given value.
• Replace the variable with its given value in every instance where it appears.
• Solve the resulting arithmetic expression.

This method is key to solving many algebraic problems efficiently and is widely applicable in varied mathematical contexts.

Order of Operations

Order of operations is a rule that ensures mathematical expressions are solved consistently and correctly.
It determines the sequence in which different operations should be performed. The common acronym PEMDAS reminds us of the correct order: Parentheses, Exponents, Multiplication, and Division (from left to right), Addition and Subtraction (from left to right).

In our exercise:

• We first perform the multiplication \(3 \times 4\), as multiplication comes before subtraction in the order of operations.
• Then, we subtract \(12\) from \(18\), which is the final operation required to evaluate the expression.

By adhering to the order of operations, you ensure that the expression is evaluated correctly. Missing or misplacing any steps can lead to errors, affecting the final outcome.

Integer Arithmetic

Integer arithmetic deals with operations involving whole numbers, which include positive numbers, negative numbers, and zero.
It's critical for solving expressions correctly, as operations like addition, subtraction, multiplication, and division follow specific rules when it comes to integers.

In the given problem:

• The multiplication \(3 \times 4\) is straightforward as both are positive integers, giving us \(12\).
• Next, the subtraction \(18 - 12\) involves two positive integers, resulting once again in a positive integer, \(6\).

Understanding integers and their properties allows you to handle signs correctly and compute expressions without errors. It's foundational for algebra and extends to more complex mathematical concepts later on.