Problem 28 - Algebra A Combined Function

Question

Simplify each expression by combining like terms. See Examples 6 through 10. $$ 14 y-30 y $$

Step-by-Step Solution

Answer

The simplified expression is \( -16y \).

Step 1: Identify Like Terms

In the given expression, identify the terms that are similar. Here, both terms involve the variable \( y \), so they are like terms.

Step 2: Combine Like Terms

To combine the like terms \( 14y - 30y \), subtract the coefficients of \( y \). The coefficients are 14 and -30.

Step 3: Calculate the Result

Subtract 30 from 14 to find the new coefficient for \( y \). \( 14 - 30 = -16 \). Thus, the simplified expression is \( -16y \).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition and subtraction. They can be as simple as a single number or variable or as complex as a polynomial with multiple terms. In the expression "14y - 30y," we see two algebraic terms, each consisting of a number (coefficient) and a variable.
The goal when manipulating algebraic expressions is often to simplify them or solve for a particular variable. This requires a solid understanding of how different parts of the expression relate to one another.
Understanding algebraic expressions is crucial because they form the basis of more complex mathematical concepts. These expressions help model real-world situations in fields ranging from physics to economics. When you tackle algebraic expressions, always keep the structure of each term in mind. Pay attention to rules such as the order of operations and how terms can be combined.

Simplification

Simplification is the process of transforming an expression into its simplest form. This makes it easier to work with and understand. The primary way we simplify expressions is by combining like terms.
For example, in the expression "14y - 30y," the objective is to consolidate these terms into a single term. Like terms have the same variable raised to the same power. In this case, both terms have the variable "y."
By simplifying expressions, calculations become more straightforward, and the risk of making errors decreases. Simplification also helps in comparing expressions and finding their values more efficiently. The process usually involves performing operations like addition or subtraction on the coefficients while keeping the variable part intact.

Coefficients

In algebra, coefficients are the numerical part of the terms in an expression. They multiply the variable they're attached to, giving weight or scale to the corresponding variable. For instance, in the term "14y," 14 is the coefficient.
Coefficients tell you how many times the accompanying variable is used, or in a real-world context, how strongly that variable affects the expression's outcome.

• If the coefficient is positive, it adds value to the expression.
• If it's negative, it decreases the value.

In our example "14y - 30y," the coefficients are 14 and -30. Simplifying this involves subtracting 30 from 14, resulting in a new coefficient of -16 for the variable 'y'. So the simplified term is "-16y." Paying attention to coefficients is crucial for correctly simplifying expressions or translating algebraic language into real-world terms.

Variables

Variables represent unknown or changeable values within a mathematical expression or equation. They are typically denoted by letters such as 'x', 'y', or 'z'. These symbols stand in for numbers that can vary or are yet to be determined.
In our example "14y - 30y," the variable is 'y'. It's a placeholder that allows the expression to be flexible, applicable across various situations.
The beauty of variables lies in their ability to generalize mathematical principles. By understanding variables and how they work within expressions, you can solve equations and model real-world problems. Always remember that while variables represent unknowns, their behavior is governed by the rules of algebra, such as combining like terms, factoring, and distributing terms across parentheses.