Problem 44 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the expression. \(4 \times 3+18 x \div 9-12\)

Step-by-Step Solution

Answer

The expression simplifies to 2.

Step 1: Evaluate Multiplication

First, we simplify the multiplication part of the expression. Multiply 4 by 3. \[ 4 \times 3 = 12 \]

Step 2: Evaluate Division

Next, we simplify the division part of the expression. Divide 18 by 9. \[ \frac{18}{9} = 2 \]

Step 3: Substitute Results and Simplify

Substitute the results of multiplication and division back into the expression and simplify it. Our expression now is:\[ 12 + 2 - 12 \] Combine like terms:\[ 12 + 2 = 14 \]Finally, subtract 12:\[ 14 - 12 = 2 \]

Key Concepts

Order of Operations

When dealing with algebraic expressions, the order of operations is crucial. This set of rules ensures consistency when simplifying expressions or equations by dictating the sequence in which operations should be carried out. Commonly remembered through the acronym PEMDAS:

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

In the expression given, following the correct order is key. We prioritize multiplication and division before looking at addition and subtraction. This hierarchy ensures everyone evaluates expressions the same way, preventing errors and misunderstandings. Remember, if multiplication or division appears first from left to right, it is completed first, similar for addition and subtraction.

Expression Evaluation

Evaluating an expression involves replacing variables with numerical values (if there are any) and simplifying the outcome using the order of operations. In this step-by-step process, it’s about breaking down complex expressions into more manageable pieces.
For our example, there are no variables involved, but each arithmetic operation is carried out in a specific order:

• First, perform multiplication: 4 multiplied by 3 gives 12.
• Next, carry out division: 18 divided by 9 results in 2.
• Finally, substitute these results back into the expression to simplify further by combining the numbers using addition and subtraction.

Being methodical and patient while evaluating expressions reduces mistakes and leads to the correct answer.

Basic Arithmetic Operations

To understand algebra, a solid grasp of basic arithmetic operations can make a big difference:

• Addition: Combining two or more numbers to achieve a total.
• Subtraction: Removing one number from another, finding the difference.
• Multiplication: Think of it as repeated addition. Multiplying is combining equal groups.
• Division: Splitting a number into equal parts or groups, essentially the opposite of multiplication.

By practicing these operations and understanding how to manipulate them, simplifying expressions can become more intuitive. In our example, the process was straightforward, thanks to knowing that multiplication and division have priority and basic arithmetic provides a foundation for interpreting results correctly.