Problem 87 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(77-96,\) simplify each algebraic expression. $$-y+4 y$$

Step-by-Step Solution

Answer

The simplified form of the algebraic expression -y + 4y is 3y.

Step 1: Identify the terms to be combined

Look at the given expression and identify the terms which can be combined. Here, the terms are -y and 4y.

Step 2: Combine similar terms

Combine the terms with common variable 'y'. The operation between them his addition/subtraction according to the sign with the variable. Here, it is -y + 4y = (-1+4)y.

Step 3: Perform operation

Perform the operation between coefficients -1 and 4. -1 + 4 equals 3. Therefore, it becomes 3y.

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one of the fundamental techniques involves combining like terms. But what exactly does this mean? Like terms are terms that have exactly the same variable parts, which means they have the same base variables raised to the same power. For instance, in the example \( -y + 4y \), both terms have the variable 'y' and thus are considered like terms.

To simplify an expression with like terms, you add or subtract the coefficients of these terms while keeping the variable part unchanged. Imagine it like grouping similar objects: if you have 1 apple and someone gives you 4 more apples, you now have 5 apples. Similarly, \( -y + 4y \) simplifies to \( 3y \) because you combine the coefficients \( -1 \) and \( 4 \) to get \( 3 \) and then reattach the variable part, 'y'.

It's essential to pay attention to the signs of the coefficients as they determine whether you add or subtract the numbers. In the given problem, \( -y \) is the same as \( -1y \) and when you add \( 4y \) to it, you're essentially adding -1 and 4, which leads to 3, hence \( 3y \) as the simplified expression.

Algebraic Terms Operation

Working with algebraic terms involves operations such as addition, subtraction, multiplication, and division. When dealing with addition and subtraction, as seen in our exercise, it's important to only operate on like terms. The operation on algebraic terms is straightforward: combine the coefficients and leave the variable unchanged if the operations are addition or subtraction.

However, things can get more complex with multiplication and division. When you multiply terms, you multiply the coefficients and apply the laws of exponents to the variables. For division, you divide the coefficients and subtract the exponents of the variables if they are similar. Always remember that only like terms can be added or subtracted, but any terms can be multiplied or divided, as long as you obey the rules for dealing with powers and variable expressions.

Example with Different Operations

Consider the terms \(2a^2b \) and \(3ab^3\). If we multiply these, we get \(6a^3b^4\), but they cannot be added or subtracted since they are not like terms.

Coefficient Operations

The coefficients in algebra are the numerical parts of the terms. They play a crucial role as they determine the value and behavior of terms when they are combined or manipulated. Coefficient operations follow the same arithmetic rules as regular numbers, meaning they can be added, subtracted, multiplied, and divided.

In the context of our example \( -y + 4y \), we see coefficient operations in action when we combine \( -1 \) and \( 4 \) during the process of combining like terms. Here, it's essential to recognize that the variable 'y' doesn't affect the operation of the coefficients - it's treated independently until the end of the calculation.

Understanding how to manipulate coefficients effectively is a key skill in algebra as it allows you to simplify expressions, solve equations, and work with polynomials. Keeping track of the signs and using the order of operations correctly will ensure accuracy in simplifying algebraic expressions.