Problem 114 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ -v-3 v+6 v+2 v $$

Step-by-Step Solution

Answer

The simplified expression is \(4v\).

Step 1: Identify like terms

The given expression is \(-v - 3v + 6v + 2v\). To simplify, first identify like terms. In this case, all terms are like terms because they all have the variable \(v\).

Step 2: Combine like terms

Combine the coefficients of the like terms. The coefficients are: \(-1, -3, 6,\) and \(2\). Add these coefficients together: \(-1 - 3 + 6 + 2 = 4\).

Step 3: Write the simplified expression

Multiply the result of the addition by \(v\) to get the simplified expression: \(4v\).

Key Concepts

Understanding Like Terms

In algebra, like terms are terms that have the same variable raised to the same power. They look alike and can be combined to simplify an expression. It’s important to note that coefficients, or the numbers in front of the variable, don’t need to be the same for terms to be considered like terms. What matters is the variable and its exponent.

In the expression

• \(-v - 3v + 6v + 2v\),

each term includes the variable \(v\), and since they all share this same variable, we recognize them as like terms. Recognizing like terms is a crucial step before performing operations such as addition or subtraction. This ensures all the terms are handled appropriately and that the math performed on them will be correct.

Coefficients in Algebra

Coefficients are the numerical parts of a term that are placed in front of a variable. In the expression we're discussing, they are the numbers directly associated with the variable \(v\). Understanding coefficients is vital because they tell us how many of each term we have to consider when combining like terms.

Consider the expression

• \(-v - 3v + 6v + 2v\),

the coefficients are -1 for \(-v\), -3 for \(-3v\), 6 for \(+6v\), and 2 for \(+2v\). Even when a number isn’t explicitly displayed, as in \(-v\), a coefficient of 1 or -1 is implied. Understanding these coefficients allows us to add or subtract them to simplify any expression accurately.

Combining Like Terms

Combining like terms is a way to simplify expressions in algebra by adding or subtracting coefficients of like terms. When you identify like terms, you use their coefficients to create a simpler expression. This makes algebraic expressions easier to handle and evaluate.

For the expression

• \(-v - 3v + 6v + 2v\),

you combine the coefficients: -1, -3, 6, and 2. Performing the arithmetic, you get

• \(-1 - 3 + 6 + 2 = 4\).

This means the simplified expression is \(4v\). By performing these operations, you reduce the complexity of the expression while retaining its value.