Problem 51 - Prealgebra

Question

Combine like terms. $$25 y+3 y-y$$

Step-by-Step Solution

Answer

The expression simplifies to \(27y\).

Step 1: Identify Like Terms

In this expression, the like terms are the terms with the variable 'y'. These terms are: \(25y\), \(3y\), and \(-y\). Like terms have the same variable raised to the same power, and can be combined to simplify the expression.

Step 2: Add and Subtract the Coefficients

To combine the like terms, add and subtract their coefficients. The expression is \(25y + 3y - y\). Here, the coefficients are 25, 3, and -1 (since \(-y\) is the same as \(-1y\)). Add 25 and 3 to get 28, and then subtract 1 to get 27.

Step 3: Write the Simplified Expression

Combine the calculated coefficients with the common variable 'y'. The result is \(27y\), which is the simplified expression when all like terms are combined.

Key Concepts

Prealgebra Concepts

Prealgebra is the foundation of algebra and mathematics as a whole. It introduces students to the basics of handling numbers and variables.

One key idea in prealgebra is understanding the concept of "terms" in an expression. A term can be a number, a variable, or both multiplied together. In 25y + 3y - y, each part of the expression represents a term.

• 25y is a term composed of a coefficient, 25, and the variable y.
• 3y follows the same pattern.
• y is actually -1y, demonstrating how subtraction in an expression can be turned into adding a negative term.

These terms can be organized and simplified, laying the groundwork for more advanced topics.

Simplifying Algebraic Expressions

Simplifying an algebraic expression involves combining like terms to make calculations easier and more efficient. This practice not only simplifies the expression but also clarifies its meaning.

In expressions like 25y + 3y - y, we need to focus on the process of combining like terms. Here are the basic steps:

• Identify like terms: These are terms that have the same variable part. In this case, all terms are like terms since they contain 'y'.
• Adjust the coefficients: Add or subtract the numbers in front of the variables. For our example, you would calculate 25 + 3 - 1 to arrive at 27.
• Rewrite: Once the terms are combined, refocus the expression into its simplest form, which is 27y.

Despite seeming straightforward, understanding how to simplify expressions paves the way for solving equations and other algebraic skills.

Variables and Coefficients

In algebraic expressions, variables and coefficients play crucial roles. Recognizing their functions can help simplify math problems. A variable represents an unknown or a number that can change. In our expression, y is a variable.

Coefficients are the numbers in front of the variables, indicating how many times the variable is considered within the expression. For example, the coefficient in 25y is 25, which tells us there are 25 'y's. Similarly, in 3y and -y, the coefficients are 3 and -1, respectively.

• Variables can be thought of as placeholders for numbers.
• Coefficients show the multiplication factor for the variable.
• Knowing these roles makes it easier to follow and apply mathematical rules.

Combining knowledge of coefficients and variables is essential in algebra for constructing and deconstructing expressions accurately.