Problem 16 - Algebra A Combined Function

Question

Simplify each expression by combining any like terms. $$ 5 y-14+7 y-20 y $$

Step-by-Step Solution

Answer

The simplified expression is \(-8y - 14\).

Step 1: Identify Like Terms

First, identify terms that consist of the same variable. In this expression, the terms with "y" are the like terms: \(5y\), \(7y\), and \(-20y\).

Step 2: Combine the Like Terms

Add and subtract the coefficients of the like terms: \(5y + 7y - 20y\). This simplifies to \((5 + 7 - 20)y = -8y\).

Step 3: Simplify Constant Terms

The expression also has a constant term, \(-14\). Since there are no other constant terms, this value remains as it is in the expression.

Step 4: Write the Simplified Expression

Combine the simplified terms to write the final expression. The expression is now \(-8y - 14\).

Key Concepts

Understanding Like Terms

Like terms are terms within an algebraic expression that contain the same variable raised to the same power. Identifying them is a fundamental step in the simplification process. For example, in the expression \(5y + 7y - 20y - 14\), the terms \(5y\), \(7y\), and \(-20y\) are considered like terms because they all involve the variable "y" raised to the first power. It's important to recognize that while these terms are like terms, \(-14\) is not a like term because it lacks the variable "y".
Breaking down like terms can help you simplify the expression by focusing on the variable parts separately from constant numbers. Once you have identified all like terms in an expression, you can move on to combining them. When doing so, focus only on their coefficients—the numerical values in front of the variables. This is crucial for performing operations such as addition or subtraction on these terms.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves reducing an expression to its simplest form. After identifying like terms, the next step is to combine them. In our example, the expression includes the like terms \(5y + 7y - 20y\). To simplify, add and subtract the coefficients:

• Add: \(5 + 7 = 12\)
• Subtract: \(12 - 20 = -8\)

This gives us \(-8y\).
Don't forget the constant term, which is any number without a variable. Here it's \(-14\). You simply keep it as part of your expression since there's no other constant term to combine it with.
The final simple expression is \(-8y - 14\). Not only does simplifying make the expression cleaner, it often reveals relationships within equations that were not immediately apparent in their longer forms.

The Role of Coefficients

Coefficients are the numbers in front of the variables in an algebraic expression. They tell us how many units of the variable we have. For instance, in \(5y\), the coefficient "5" indicates there are five units of "y." Properly managing coefficients is key to combining like terms.
To simplify an expression like \(5y + 7y - 20y\), focus on the coefficients. Add or subtract them while ignoring the variables themselves during these calculations:

• Start with: \(5 + 7\) gives \(12\)
• Then subtract: \(12 - 20\) equals \(-8\)

Thus, the resulting term \(-8y\) shows that you have a negative eight units of "y."
Preservation of coefficients is crucial in maintaining the correct proportion and direction (positive or negative) of each term—and thereby the entire expression—through manipulations like simplification.