Problem 98 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ -16 y+16 y $$

Step-by-Step Solution

Answer

The expression simplifies to 0.

Step 1: Identify Like Terms

Observe that in the expression \(-16y + 16y\), both terms contain the variable \(y\) and thus are like terms. Like terms have the same variables raised to the same power.

Step 2: Combine Like Terms

Since \(-16y\) and \(16y\) are like terms, we can combine them by adding their coefficients. This involves performing the arithmetic operation \(-16 + 16 = 0\).

Step 3: Simplify the Expression

Substitute the result from the previous step back into the expression. Since the sum of the coefficients is zero, the expression simplifies to \(0 \cdot y = 0\).

Key Concepts

Understanding Like Terms

When you're working with algebraic expressions, identifying like terms is crucial. But what exactly are they? Like terms are terms that have the same variable(s) raised to the exact same power. For example, in the expression

• \(-16y\)
• \(16y\)

both terms have the variable \(y\). So they're like terms.
Whether the coefficient (the number in front) is positive or negative, terms that share these characteristics can be grouped together.
To be like terms, they must have:

• The same variable
• The same exponent

Recognizing these is a fundamental skill to master before moving on to more complex operations in algebra.

Combining Like Terms

Once you have identified like terms in an expression, the next step is to combine them. But why should we do this?
Combining like terms simplifies expressions, making them easier to work with. It involves adding or subtracting the coefficients — the numerical part of the terms.For instance, in the expression \(-16y + 16y\), once you identify them as like terms, you can combine by adding the coefficients:

• -16 (from the first term) + 16 (from the second term)

This calculation gives us 0.
Combining like terms doesn't change the value of the expression; it simplifies it into a more workable form.
This technique is particularly helpful when solving equations.

Simplifying Expressions

Simplifying expressions is the process of condensing them into their simplest form. After combining like terms, you will often find that further simplification is possible by evaluating any numerical calculations. For example, in the expression

• \(-16y + 16y = 0\)

you see that the terms cancel out each other, reducing the expression to 0.Simplification helps in understanding and solving algebraic equations more efficiently. It can involve:

• Combining like terms
• Adding or subtracting coefficients
• Performing arithmetic calculations

Each step helps to declutter the expression and allow a clearer view of its components.
Simplifying ensures that your final expression is as neat and as simple as it can be, aiding in ease of interpretation and further manipulation.