Problem 84 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. \(-6 x-9 y-4 x+15 y\)

Step-by-Step Solution

Answer

The simplification of the given algebraic expression results in \( -10x + 6y \)

Step 1: Identify Like Terms

In this expression, the like terms are terms having the same variable parts. Therefore, -6x and -4x are like terms; -9y and 15y are like terms.

Step 2: Combine Like Terms

Add or subtract the coefficients of like terms. So in this case, combine -6x and -4x to get -10x. Also combine -9y and 15y to get 6y.

Step 3: Write the Simplified Expression

Express the simplified form as -10x + 6y.

Key Concepts

Like Terms in Algebra

Understanding 'like terms' is critical to mastering algebra and improving problem-solving efficiency. Like terms in algebra are terms that have identical variables raised to the same power. For example, in the terms 3a and 5a, both have the variable 'a', which makes them like terms. However, 3a and 3b are not like terms because they contain different variables.

Additionally, the exponents on the variables must match for terms to be considered 'like.' Therefore, while 2x and -5x are like terms, 2x and 2x^2 are not, because the latter includes a squared variable. Consistently recognizing like terms makes combining them straightforward, ultimately guiding you towards the simplified form of an algebraic expression.

Combining Like Terms

Once you've identified like terms in an algebraic expression, the next step is to combine them. This process, known as combining like terms, is essentially an exercise in basic arithmetic, where you add or subtract the coefficients of like terms. The variables remain unchanged. Consider the expression from our example, \( -6x - 4x \). Both terms have an 'x' variable, making them like terms. To combine them, simply add their coefficients: \( -6 + (-4) \), which equals \( -10 \). Hence, their combination is \( -10x \).

By performing this action for all like terms in an expression, you simplify the problem immensely, making it more manageable and cleaner for further operations or evaluations. Be vigilant about the signs in front of the coefficients as well; they determine whether you're adding or subtracting the values.

Simplified Form in Algebra

The ultimate goal when dealing with algebraic expressions is to reach their simplest form. This does not necessarily mean the shortest expression, but rather the most straightforward version without redundancies. Simplifying an algebraic expression involves reducing it by combining like terms, using the distributive property if necessary, and performing any arithmetic operations to eliminate complex fractions or nested parentheses.

In the context of our example, we would consider the expression \( -6x - 9y - 4x + 15y \) simplified once we've combined the like terms of \(x\) and \(y\) separately. After this step is completed, the simplified expression is \( -10x + 6y \), which no longer has redundant terms. It's worth mentioning that a simplified expression will also make solving for variables, should that be required, a much less daunting task.