Problem 34 - Algebra 1: Concepts and Skills

Question

Simplify the expression. \(-2(k)\)

Step-by-Step Solution

Answer

The simplified expression is \(-2k\).

Step 1: Identify the Expression

The given expression to simplify is \(-2(k)\).

Step 2: Simplify Multiplication

\(-2(k)\) is the same as \(-2 * k\). Multiply the \(-2\) with the \(k\) which results in \(-2k\).

Key Concepts

Multiplication

Multiplication in algebra is a fundamental operation that involves finding the product of two or more numbers or variables. When multiplying algebraic terms, you simply multiply the coefficients and then write down the variables side by side, retaining their original order and any exponents.

For instance, in the expression \(-2(k)\), you are asked to multiply \(-2\) by the variable \k\. This is written as \(-2 * k\). There's no exponent attached to \(k\) here, so you just attach the \k\ to the multiplying coefficient \(-2\) to get \(-2k\).

Remember:

• Align coefficients and variables: Multiply the numbers first and then write the variable(s).
• Keep the variables with their exponents: If there were exponents, you'd deal with those as another separate step.
• Understand the operation: In algebra, combining like terms through multiplication keeps expressions neat and manageable.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are an essential part of algebra which allows us to represent mathematical situations symbolically.

For example, expressions like \(2x + 3\), \(-y + 4\), or \(-2k\) show how numbers and variables can be connected using arithmetic operations such as addition, subtraction, and multiplication.

Key aspects of working with algebraic expressions include:

• Simplifying: Combining like terms and using basic arithmetic operations to make the expression easier to work with.
• Substituting: Replacing variables with numbers to evaluate the expression.
• Translating: Articulating real-world problems as symbolic expressions.

In the exercise given, the expression \(-2(k)\) represents a simple multiplication of a number with a variable, resulting in a product that maintains all the properties of algebraic terms.

Negative Numbers

Negative numbers are numbers less than zero, important in both simple arithmetic and more complex algebraic problem-solving. They are represented with a minus sign \-\ placed before them. Understanding how negative numbers interact in mathematical operations is essential, particularly in multiplication.

When you multiply a negative number by a positive number, the result is also a negative number. For example, multiplying \(-2\) by \k\ in \(-2(k)\) results in \(-2k\). This is because, by rule, a positive times a negative or a negative times a positive always results in a negative.

Key points to remember about negative numbers in multiplication include:

• A negative multiplied by a positive gives a negative result.
• A negative multiplied by another negative gives a positive result.
• Recognizing signs is crucial in applying these rules correctly.

Familiarity with these rules will help you move smoothly between different types of mathematical problems, particularly those involving algebraic expressions containing negative coefficients or factors.