Problem 54 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$5 y+(-11 z)+(-15 y)+20 z$$

Step-by-Step Solution

Answer

The simplified version of the algebraic expression is \(-10y + 9z\).

Step 1: Identify and Combine the Like Terms

Combine the terms with the variable \(y\): \(5y - 15y\) and terms with the variable \(z\): \(-11z + 20z\).

Step 2: Simplification of Step 1

The previous step gives \(-10y + 9z\) as the simplified version of the original expression. This cannot be simplified any further as the terms are not like terms.

Key Concepts

Combining Like Terms

In algebra, combining like terms is a fundamental process for simplifying expressions. Like terms are terms that have the same variables raised to the same power. For example, in the expression 5y + (-11z) + (-15y) + 20z, the terms with the variable y are 5y and -15y, and they can be combined because they have the identical variable y. Similarly, the terms -11z and 20z are like terms because both contain the variable z.

To combine these like terms, we simply perform the arithmetic operation indicated:

• For y: 5y - 15y results in -10y.
• For z: -11z + 20z results in 9z.

As a result, the expression simplifies to -10y + 9z. Remember that when combining like terms, only the coefficients (numerical factors) are added or subtracted, while the variable part remains unchanged.

Algebraic Simplification

Algebraic simplification is the process of making an algebraic expression as simple as possible. This involves combining like terms, as we did with the expression 5y + (-11z) + (-15y) + 20z, by adding or subtracting the coefficients of the like terms. This step is essential for solving equations or making the expressions easier to understand and work with. After combining like terms in the given exercise, the simplified form is -10y + 9z. At this point, the expression cannot be simplified further because y and z are not like terms and thus cannot be combined.

Important tips for algebraic simplification:

• Ensure that you only combine terms with the exact same variable parts.
• Always bring similar terms together before performing the arithmetic operations.
• Keep track of positive and negative signs as they can change the result.

Intermediate Algebra

Intermediate algebra covers a broad range of topics beyond elementary algebra, providing a deeper understanding of equations, inequalities, and functions. Simplifying algebraic expressions, as done in this exercise, lays the groundwork for more complex operations such as factoring polynomials, solving quadratic equations, and understanding rational expressions. Mastery of combining like terms and performing algebraic simplification is key to success in intermediate algebra.

As students advance, they encounter expressions with more variables and more complex coefficients. It is important to approach these systematically, always remembering the basic principles of algebra, including the distributive property, the associative property, and the commutative property. These principles help to restructure and simplify expressions, preparing them for further manipulation in more advanced problem-solving scenarios.