Problem 44 - College Algebra

Question

For the following exercises, simplify the expression. $$ 4 \cdot 3+18 x \div 9-12 $$

Step-by-Step Solution

Answer

The simplified expression is \(2x\).

Step 1: Follow the Order of Operations

The order of operations is parentheses, exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right). Begin by performing the multiplication and division from left to right.

Step 2: Multiply 4 by 3

Calculate the first multiplication in the expression: \[4 \cdot 3 = 12\] Thus, the expression updates to: \[12 + 18x \div 9 - 12\]

Step 3: Divide 18x by 9

Perform the division operation: \[18x \div 9 = 2x\] The expression is now: \[12 + 2x - 12\]

Step 4: Simplify by Addition and Subtraction

Combine like terms by performing the addition and the subtraction:First, calculate: \[12 - 12 = 0\] Then, update the expression: \[0 + 2x = 2x\]

Key Concepts

Simplifying Expressions

Simplifying expressions is one of the foundational skills in algebra. It's about reducing an expression to its simplest form while ensuring its original value remains unchanged. This process involves applying the order of operations correctly to combine like terms, perform operations, and reduce the expression to a more manageable form.

To simplify effectively:

• Identify and perform operations like multiplication, division, addition, and subtraction as dictated by the order of operations.
• Combine like terms, which means gathering similar terms like all the numbers together or all terms containing the same variable.
• Keep expressions neat and aligned to avoid mistakes, which is especially important when dealing with longer expressions.

Consider the expression from our example: \[4 \cdot 3+18x \div 9-12\].Initially, it might seem complex, but following these steps makes it straightforward and simple.

Multiplication and Division

Multiplication and Division are critical components of arithmetic and algebra, and they often appear in expressions we need to simplify. According to the order of operations, multiplication and division are to be performed after parenthesis but before addition and subtraction.

Let's examine how these operations are handled in the expression:

• \(4 \cdot 3\) is the first multiplication encountered, calculated as \(12\).
• For the division, \(18x \div 9\) simplifies to \(2x\). This step reduces the complexity of the quotient involving a variable.
• Remember, multiplication and division should be performed from left to right as they appear in the expression.

By systematically performing multiplication and division before moving on to addition and subtraction, the expression becomes increasingly manageable, setting the stage for the final simplification steps.

Addition and Subtraction

Addition and subtraction are the last steps in the order of operations, and it's important to follow this order to achieve a correctly simplified expression. The key to effectively handling these operations lies in combining like terms and ensuring accurate calculation.

Looking at our simplified expression: \[12 + 2x - 12\], we:

• First, perform the subtraction \(12 - 12\), which results in \(0\).
• This reduces the expression to \(0 + 2x\).
• Finally, simplify to get \(2x\), which is the simplest form of the original expression.

Always remember that when dealing with addition and subtraction, combining like terms efficiently not only simplifies the expression but also ensures it accurately represents its algebraic relationships. This approach results in a clean, final expression that captures the essence of the original complexity.