Problem 3 - College Algebra with Modeling and Visualization

Question

Combine like terms whenever possible. $$5 y^{7}-8 y^{7}$$

Step-by-Step Solution

Answer

The combined term is \(-3y^7\).

Step 1: Identify Like Terms

Identify the terms in the expression that have the same variable part. In this case, both terms, \(5y^7\) and \(-8y^7\), have \(y^7\) as the variable part, meaning they are like terms.

Step 2: Combine Like Terms

Add the coefficients of the like terms. The expression is \(5y^7 - 8y^7\). Calculate \(5 - 8\), which results in \(-3\). So, the combined term is \(-3y^7\).

Key Concepts

Like Terms

In algebra, "like terms" are terms that contain the same variable raised to the same power.
In the expression from the exercise, both terms, \(5y^7\) and \(-8y^7\), have the same variable, \(y\), raised to the same power, \(7\).
This commonality allows us to combine them easily since they are like terms. Understanding this concept is essential:

• Like terms can be combined using addition or subtraction of their coefficients.
• Only terms that have the exact same variable factor can be added or subtracted like this.
• Examples of like terms: \(3x^2\) and \(6x^2\), \(-y\) and \(4y\).

Recognizing like terms simplifies expression manipulation, making calculations more straightforward.

Coefficients

In an algebraic expression, the "coefficient" is the number that is multiplied by the variable. It provides a multiplier for the variable part of the term. For example, in the expression \(5y^7\), the coefficient is \(5\), while in \(-8y^7\), the coefficient is \(-8\).
Understanding coefficients is crucial in algebra:

• Coefficients indicate how many times a term is counted in the expression.
• They are the numbers that you add or subtract when combining like terms.
• Positive and negative signs associated with coefficients indicate addition or subtraction in the context of combined like terms.

In the exercise, you subtract these coefficients \(5 - 8\) to get \(-3\), which becomes the new coefficient when combining the terms into \(-3y^7\).

Variables

"Variables" are symbols used in algebra to represent numbers. They give expressions their flexibility, allowing you to express general mathematical relationships.
In our example, the variable is \(y\).
Variables are essential because:

• They allow for the generalization of relationships in algebra.
• Variables can represent unknowns or changeable values in an expression.
• When working with variables, you apply the same operations across like terms.

In expressions like \(5y^7\), the variable \(y\) is consistent across terms, which helps determine the "likeness" of terms. This constancy means you can combine their coefficients effectively.