Problem 35 - Elementary Algebra

Question

Simplify. $$ 13-12 \cdot 15 $$

Step-by-Step Solution

Answer

The simplified expression is -167.

Step 1: Recognize Order of Operations

To simplify the expression \(13 - 12 \cdot 15\), we must apply the order of operations, which follows the PEMDAS/BODMAS rules: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). In this expression, we first perform the multiplication.

Step 2: Perform Multiplication

Multiply \(12\) by \(15\): \[12 \cdot 15 = 180\]

Step 3: Subtract the Product from 13

Subtract \(180\) from \(13\): \[13 - 180 = -167\]

Step 4: Conclusion

The simplified result of the expression \(13 - 12 \cdot 15\) is \(-167\).

Key Concepts

PEMDAS

PEMDAS is an acronym used to remember the order of operations in evaluating expressions. It stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This sequence ensures that everyone calculates expressions in a consistent manner.

To understand PEMDAS better:

• Parentheses: Always address expressions inside parentheses first. This applies even if there are multiple layers of parentheses.
• Exponents: Solve numbers with exponents next, which are powers or roots of numbers.
• Multiplication and Division: These operations are performed next. They are on the same level, so proceed from left to right.
• Addition and Subtraction: Lastly, these are the simplest operations, also done from left to right.

By using PEMDAS, complex expressions like the one in our exercise becomes much more manageable and understandable.

Simplification

Simplification is the process of transforming an expression into its simplest form. This involves reducing or combining terms while maintaining the expression's original value. In our example, we performed simplification as we moved through different steps following PEMDAS.

When simplifying:

• Ensure to follow the order of operations diligently to avoid mathematical errors.
• Look for opportunities to combine like terms, such as similar coefficients or variables.
• Be consistent - ensure each step maintains the equality of the original expression.

This step-by-step method allows expressions to be solved efficiently and accurately, reducing errors in mathematical processes. By practicing these techniques, you'll develop a stronger understanding of how algebraic expressions work.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. Simplifying them involves adhering to mathematical rules to find their simplest form without changing their values. This can involve using PEMDAS to ensure each part is correctly addressed.

Some key elements of algebraic expressions are:

• Terms: These are parts of an expression separated by addition or subtraction signs.
• Coefficients: These are numerical values multiplying the variables within each term.
• Operations: Arithmetic operations in expressions tell how to combine numbers and variables.

Understanding algebraic expressions helps in solving equations, forms the basis of algebra, and is critical for advanced mathematics. It combines logic and computation to bring clarity to mathematical problems.