Problem 113 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ -4 r-7 r+2 r-r $$

Step-by-Step Solution

Answer

The simplified expression is \(-10r\).

Step 1: Identify Like Terms

First, we need to identify all the like terms in the expression. Like terms are terms that have the same variable raised to the same power. In the expression \[-4r - 7r + 2r - r,\]all the terms are like terms because they all contain the variable \(r\) raised to the first power.

Step 2: Combine Like Terms

Now, we will combine the like terms by adding or subtracting their coefficients. The expression is \[-4r - 7r + 2r - r.\]To combine these, we simply add or subtract the coefficients:\[(-4 - 7 + 2 - 1)r.\]

Step 3: Simplify the Coefficients

Calculate the sum of the coefficients:\[-4 - 7 + 2 - 1 = -10.\]

Step 4: Rewrite the Expression

Using the simplified coefficients, rewrite the expression as:\[-10r.\]

Key Concepts

Understanding Like Terms

In algebraic expressions, recognizing like terms is crucial. Like terms are the building blocks that allow us to simplify expressions effectively. They are terms that have identical variable parts. In other words, the variables must be the same and raised to the same power. For example, in the expression

• -4r
• -7r
• 2r
• -r

each term contains the same variable 'r' raised to the first power. Since the variable and its power (r¹) match across each term, these are all like terms. Being able to identify these terms is the first step towards simplifying any algebraic expression.

Combining Like Terms

After identifying like terms, the next step is to combine them. This involves adding or subtracting their coefficients. The coefficient is the numerical part of the term that is multiplied by the variable. In our example, the expression

• -4r
• -7r
• 2r
• -r

lets us know that the coefficients are -4, -7, 2, and -1 respectively.
To combine them, you simply perform the arithmetic operation on the coefficients. In this instance, the arithmetic operation is:

• (-4 - 7 + 2 - 1)

This calculation will yield the new coefficient for the simplified term. Combining like terms is an essential process in algebra that leads us toward a simplified expression.

Algebraic Simplification Process

Once we have combined the coefficients of the like terms, the task is to simplify the expression by writing it in a cleaner, simpler form. Continuing with our example:

• The combination of coefficients results in -10.
• The simplified expression then becomes -10r.

This final expression represents the same value as the original one but in a much more streamlined format. Algebraic simplification is all about clarity and ease of understanding. Each simplification step simplifies the problem further, eliminating unnecessary complexity and making the expression easier to work with for further algebraic operations or solving equations.