Problem 103 - Algebra A Combined Function

Question

Perform each indicated operation. \(9 y^{2}-8 y^{2} \quad\)

Step-by-Step Solution

Answer

The result is \(y^{2}\).

Step 1: Identify Like Terms

In the expression \(9y^{2} - 8y^{2}\), both terms are like terms because they have the same variable part \(y^{2}\). Like terms can be combined by performing the indicated operation on the coefficients.

Step 2: Subtract the Coefficients

To combine the like terms, subtract the coefficient of the second term from the coefficient of the first term: \(9 - 8 = 1\).

Step 3: Combine the Result with the Variable Part

Attach the resulting coefficient to the common variable part: \(1y^{2}\). The simplified expression is \(y^{2}\).

Key Concepts

Like Terms

Like terms in algebra are terms that contain the same variables raised to the same power. They are like peas in a pod; they match exactly in their variable composition. In the expression given in the exercise, both terms are defined as like terms because they share the variable part \( y^{2} \). This means that the terms can be combined or simplified. Simplifying such expressions becomes easier when you understand that only the coefficients (more on that later) need to be added or subtracted, as the variable parts remain unchanged. Think of it like adding apples: \( 9 \) apples minus \( 8 \) apples still leaves you with \( 1 \) apple, where the apples are the like terms here.

Coefficients

Coefficients are the numerical parts of a term, which are multiplied by the variable part. In the expression \(9y^{2} - 8y^{2}\), the coefficients are \(9\) and \(8\). Coefficients act like the managers in a business; they dictate how much of each variable part (like employees) you have in an expression. When like terms are combined, as in subtraction or addition, only the coefficients are manipulated. In our exercise, subtracting \(8\) from \(9\) gives us \(1\), which is the new coefficient for the term \(y^{2}\). It's much like dealing with numbers on their own because you're performing arithmetic on these numbers separate from the variables.

Simplification

Simplification in algebra is the process of reducing expressions to their simplest forms, making them easier to interpret and use. Think of it like tidying up a room until you can easily find what you need. In our exercise, once like terms have been identified and their coefficients adjusted as explained before, the final step is to present the result in the cleanest form: \(y^{2}\) in this case. Simplification is key because it allows us to handle complex problems with ease. Whether you are rearranging a messy expression into something more elegant, or just performing simple operations like those in this problem, the goal is to achieve clarity in mathematical expressions.