Problem 52 - Prealgebra

Question

Combine like terms. $$12 y+4 y-y$$

Step-by-Step Solution

Answer

The simplified expression is \(15y\).

Step 1: Identify Like Terms

Look at the expression to determine which terms can be combined. The expression is \(12y + 4y - y\). Each term involves \(y\), so all are like terms and can be combined.

Step 2: Combine Coefficients

Add the coefficients of \(y\) together. Start by adding 12 and 4, which gives 16. Then subtract the coefficient of the last term, which is 1. So, the calculation becomes: \(12 + 4 - 1 = 15\).

Step 3: Rewrite the Expression

Now that the coefficients have been combined, attach the common variable \(y\) to this new coefficient. This gives the simplified expression \(15y\).

Key Concepts

Prealgebra Basics

Prealgebra is a foundational branch of mathematics that prepares you for algebra. At this stage, you learn basic concepts that underlie algebraic thinking, such as working with numbers and basic arithmetic. In prealgebra, concepts focus on:

• Understanding Numbers: Recognizing and using whole numbers, fractions, and decimals.
• Recognizing Patterns: Identifying numerical patterns and sequences.
• Introducing Variables: Starting to use symbols, usually letters, to represent numbers.
• Basic Operations: Mastering the addition, subtraction, multiplication, and division of numbers and simple expressions.

These concepts build the groundwork for more complex algebraic operations, like combining like terms and solving equations.

Simplifying Expressions: Combining Like Terms

Simplifying expressions through combining like terms is a key skill in algebra and prealgebra. It helps in reducing expression complexities and making them easier to understand or solve. To combine like terms, follow these steps:

• Identify Like Terms: Like terms are terms that have the same variable raised to the same power. For example, in the expression \(12y + 4y - y\), each term contains the variable \(y\), making them like terms.
• Combine Coefficients: Only the coefficients (numbers in front of the variables) are added or subtracted. In our example: \(12 + 4 - 1\).
• Simplify: After adding the coefficients, the expression \(15y\) is the simplest form, showing that repeated terms with the same variable can be summed up to simplify.

Simplified expressions make it easier to solve equations or further manipulate algebraic expressions.

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They are foundational in algebra and are used to express mathematical relationships. Key components of algebraic expressions include:

• Variables: Symbols, often letters, that stand in for unknown values. For example, in \(12y + 4y - y\), \(y\) is the variable.
• Coefficients: Numbers that multiply the variable (e.g., 12 in \(12y\)). These are crucial as they determine the term's value given a specific variable.
• Constants: Specific numbers without variables, though not present in every expression.
• Operators: Symbols indicating operations like addition (+) or subtraction (-).

Understanding these parts helps when you're asked to simplify or manipulate expressions, making math problems easier to solve.