Problem 25 - Fundamentals of Mathematics

Question

Simplify each expression by combining like terms. $$-3 y+4 z-11-3 z-2 y+5-4(8-3)$$

Step-by-Step Solution

Answer

The simplified expression is \(-5y + z - 26\).

Step 1: Distribute and Simplify Constants

First, simplify the expression by distributing and performing basic arithmetic operations. Start with the expression: \(-3y + 4z - 11 - 3z - 2y + 5 - 4(8 - 3)\).Calculate the value inside the parenthesis:\(8 - 3 = 5\).Then apply the multiplication:\(-4 \times 5 = -20\).Substitute back to get:\(-3y + 4z - 11 - 3z - 2y + 5 - 20\).

Step 2: Collect Like Terms

Identify and group like terms in the expression:- For terms involving \(y\): - Combine \(-3y\) and \(-2y\) to get \(-5y\).- For terms involving \(z\): - Combine \(4z\) and \(-3z\) to get \(1z\) or just \(z\).- For the constant terms: - Combine \(-11\), \(+5\), and \(-20\) to get \(-26\).The expression is now:\(-5y + z - 26\).

Step 3: Write the Final Expression

The final simplified expression, after combining all like terms, is:\[-5y + z - 26\].This expression cannot be simplified further as there are no additional like terms to combine.

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, a common and crucial step is to combine like terms. Like terms are terms in an expression that contain the same variables raised to the same power. For example, in an expression containing \(3x\) and \(-5x\), both terms are like terms because they contain the variable \(x\), each raised to the same power of 1.

• Identify terms that can be combined. Look for those that have the same variable and exponent.
• Only the coefficients of like terms are combined, not the variables themselves.
• This step simplifies the expression further, making it easier to handle in calculations.

Let's consider the terms in our expression: \(-3y\) and \(-2y\) are like terms as they both have the variable \(y\). So we can simplify them to \(-5y\). Similarly, \(4z\) and \(-3z\) are like terms which combine to form \(z\). Constants like \(-11\), \(5\), and \(-20\) are also considered like terms, resulting in the combined constant \(-26\).

Distribution in Algebra

Distribution is an important method used in algebra to eliminate parentheses by applying multiplication across terms within the parenthesis. In our exercise, we encounter the term \(-4(8-3)\), which needs distribution.

• First, simplify the terms inside the parentheses. Here, \(8-3\) simplifies to \(5\).
• Next, distribute by multiplying the term outside the parentheses with each term inside. This gives \(-4 \times 5 = -20\).
• The new term \(-20\) replaces the entire expression \(-4(8-3)\).

Distribution helps to break expressions down into simpler parts, allowing for easier combination of like terms in subsequent steps.

Basic Arithmetic Operations

Arithmetic is the foundation of algebra. It involves basic operations like addition, subtraction, multiplication, and division, which are used frequently to simplify expressions. In our problem, these operations play major roles.

• Always prioritize operations within parentheses, as demonstrated by evaluating \(8 - 3\) first.
• Subsequent arithmetic involves multiplying to distribute, such as \(-4 \times 5 = -20\).
• Finally, combine constants through addition or subtraction. For instance, \(-11 + 5 - 20\) combines to \(-26\).

Each of these operations reduces the complexity of an expression. After breaking down complex expressions, combining like terms is the next step which finishes simplifying the algebraic expression.