Problem 75 - Elementary and Intermediate Algebra

Question

Simplify by combining like terms. $$ 13 r-12 r $$

Step-by-Step Solution

Answer

The simplified expression is \( r \).

Step 1: Identify Like Terms

In the expression \( 13r - 12r \), notice that both terms have the variable \( r \). Like terms are terms that have the same variable raised to the same power. Here, both terms have the same variable \( r \).

Step 2: Combine Coefficients

Since the terms are like terms, you can combine them by adding or subtracting their coefficients. For \( 13r - 12r \), the coefficients are 13 and -12, so subtract these coefficients: \( 13 - 12 = 1 \).

Step 3: Write the Simplified Expression

After combining the coefficients, the simplified expression is \( 1r \), which can be further simplified to just \( r \).

Key Concepts

Understanding Like Terms

In algebra, like terms are essential for simplifying expressions. Like terms are terms that include the same variables raised to the same power. For example, in the expression \( 13r - 12r \), both terms contain the variable \( r \), making them like terms. The coefficient or numbers in front of the variables don't matter for them to be like terms. This means \( 13x \) and \( -4x \) are like terms because they both have the variable \( x \).
By recognizing like terms, you can easily combine them for simplification, leading to cleaner expressions and easier problem-solving steps in algebra. Remember, even if the coefficients differ, as long as the variables and their powers match, they are like terms.

Combining Coefficients

Combining coefficients is a crucial step when working with like terms in algebraic expressions. Coefficients are the numbers in front of the variables. When you combine like terms, you focus on these coefficients.
Using the previous example of \( 13r - 12r \), we see the coefficients are 13 and -12. We can either add or subtract them, depending on the operation indicated in the expression. Here, we subtract the coefficients: \( 13 - 12 = 1 \).
When combining, always perform the indicated operations (addition or subtraction) to the coefficients while keeping the variable part constant. This simplifies the expression and makes it easier to handle and solve in further algebraic equations.

Simplifying Expressions

Simplifying expressions involves pulling together all the steps of identifying like terms and combining coefficients to achieve the simplest form of an expression.
After identifying like terms in an expression such as \( 13r - 12r \) and combining the coefficients to become 1, we can write the expression as \( 1r \). However, it is usually further simplified to just \( r \).
Simplified expressions mean all like terms are combined, coefficients are properly calculated, and the expression is written in an easy-to-understand form. This results in easier computation in more complex equations and makes finding solutions much simpler.