Problem 109 - Algebra A Combined Function

Question

Simplify each expression by combining any like terms. $$ y-10+y $$

Step-by-Step Solution

Answer

The simplified expression is \( 2y - 10 \).

Step 1: Identify Like Terms

In the expression \( y - 10 + y \), identify the like terms. Here, \( y \) and \( y \) are like terms since they both contain the variable \( y \). The number \( -10 \) is a constant and isn't like a variable term.

Step 2: Combine Like Terms

Add the coefficients of the like terms together. In this case, we have two \( y \) terms, so we add them: \( y + y = 2y \). The constant \( -10 \) remains as it is.

Step 3: Write the Simplified Expression

Combine the result from Step 2 to form the simplified expression: \( 2y - 10 \). This new expression is the simplest form of \( y - 10 + y \).

Key Concepts

Understanding Like Terms

Like terms are terms in an algebraic expression that have identical variable parts and corresponding powers. This means they can be easily combined for simplification. For instance, in the expression \( y - 10 + y \), both \( y \) and \( y \) are like terms because they share the same variable \( y \) without any exponents.
However, the constant \(-10\) does not have a variable attached to it, which makes it a non-like term with respect to the other \( y \) terms.
When dealing with like terms, we look at each term’s coefficient, which is the number standing in front of the variable. This number tells us how many of that particular variable term we have. Identifying like terms is the first crucial step in simplifying complex expressions.

Simplifying Expressions

Simplifying expressions involves making an algebraic expression as concise and clear as possible. This requires performing arithmetic operations and combining any like terms that appear in the expression.
Consider our example \( y - 10 + y \). We first identify the like terms using our understanding from the previous concept. Next, we perform the arithmetic operations necessary to combine these terms, if applicable.
The goal of simplifying expressions is to represent the same mathematical relationship with fewer terms. This makes it easier to evaluate, use in further equations, or understand the relationship it describes. The simplified expression is \( 2y - 10 \), which is easier to work with and interpret compared to the original expression.

Combining Like Terms

After identifying like terms, the next step is to combine them. This simplification step is what turns a longer expression into a shorter one. The process of combining involves summing or subtracting the coefficients of like terms.
In the example \( y - 10 + y \), we identified the terms \( y \) and \( y \) as like terms. Both have an implied coefficient of 1, so we simply add these coefficients together: \( 1y + 1y = 2y \). The constant \(-10\) remains unchanged because it shares no common variable part with the \( y \) terms.
Combining like terms simplifies the expression by reducing the number of terms. The end result, \( 2y - 10 \), is an expression that retains the same information as \( y - 10 + y \) but in a more manageable form.