Verified Answer :
- Define "like terms" as terms having the same variables raised to the same powers, independent of their coefficients.
- Compare each item from the list to the target expression \(6y\), which requires the presence of only the variable \(y\) raised to the first power.
- Identify the matching items: \(3y, y, 4y, -11y,\) and \(-y\).
- Conclude that the like terms to \(6y\) are \(\boxed{3y, y, 4y, -11y, -y}\).
Explanation
- Analyzing the definition of like terms
To identify the like terms of the expression \(6y\), we must first understand what "like terms" are. In algebra, two or more terms are considered "like terms" if they contain the same variables raised to the same powers, regardless of their numerical coefficients. Our target term is \(6y\), which can be written as \(6 \cdot y^1\). This means that any term containing only the variable \(y\) raised to the first power is a like term to \(6y\). We will examine each given option to determine if it meets this criterion.
- Evaluating the given list of terms
We evaluate each item in the list against our criteria (containing only \(y\) to the first power):
- \(3y\): This contains only the variable \(y\) raised to the first power. It is a like term.
- \(5x\): This contains the variable \(x\). It is not a like term.
- \(6xy\): This contains the variables \(x\) and \(y\). It is not a like term.
- \(-2x\): This contains the variable \(x\). It is not a like term.
- \(y\): This can be written as \(1 \cdot y\). It contains only the variable \(y\) raised to the first power. It is a like term.
- \(4y\): This contains only the variable \(y\) raised to the first power. It is a like term.
- \(6\): This is a constant and contains no variable. It is not a like term.
- \(-11y\): This contains only the variable \(y\) raised to the first power. It is a like term.
- \(x\): This contains the variable \(x\). It is not a like term.
- \(-y\): This can be written as \(-1 \cdot y\). It contains only the variable \(y\) raised to the first power. It is a like term.
- Conclusion and identifying the final like terms
Based on our evaluation, the terms that are like terms to \(6y\) are those that consist only of the variable \(y\) raised to the first power. These terms are:
\(3y\), \(y\), \(4y\), \(-11y\), and \(-y\).
The final set of identified like terms is \(\boxed{3y, y, 4y, -11y, -y}\).
Examples
Identifying like terms is essential when you need to simplify algebraic expressions. For example, if you are calculating the total cost of purchasing several items where some are priced by weight (represented by \(y\)) and others by unit (represented by \(x\)), being able to group like terms allows you to combine your costs correctly. If you have costs of \(3y\), \(5x\), \(4y\), and \(x\), you group the \(y\)-terms together \((3y + 4y = 7y)\) and the \(x\)-terms together \((5x + x = 6x)\) to simplify your total cost to \(7y + 6x\).