Problem 23 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ 16-1 x 21 x-4 $$

Step-by-Step Solution

Answer

The simplified expression is \(12 - 21x\).

Step 1: Express the Problem Mathematically

First, we need to understand the expression to be simplified: \(16 - 1 \times 21 \times x - 4\). It represents the calculation order as per arithmetic operations. Remember that multiplication takes precedence over addition and subtraction.

Step 2: Perform the Multiplication

Given the expression \(16 - 1 \times 21 \times x - 4\), compute the multiplication: \(1 \times 21 \times x = 21x\). This simplifies the expression to \(16 - 21x - 4\).

Step 3: Simplify by Combining Like Terms

Now, we simplify the expression by combining the constant terms: \(16 - 4 = 12\). The expression becomes \(12 - 21x\).

Step 4: Write the Final Simplified Expression

The simplified expression of the original is \(12 - 21x\).

Key Concepts

Order of Operations

When dealing with algebraic expressions, understanding the order in which to perform operations is crucial. This principle is often summarized by the acronym PEMDAS or BODMAS. Each letter stands for:

• P/B - Parentheses/Brackets: Solve expressions inside parentheses or brackets first.
• E/O - Exponents/Orders: Next, handle any exponents or powers.
• MD - Multiplication and Division: From left to right, execute these third.
• AS - Addition and Subtraction: Lastly, carry out addition and subtraction from left to right.

In the given exercise, observe that multiplication (\(1 \times 21 \times x\)) takes precedence over subtraction and addition. By following the correct sequence, the expression was effectively simplified.

Arithmetic Operations

Arithmetic operations such as addition, subtraction, multiplication, and division form the backbone of simplifying expressions. Each operation serves a unique purpose and must be applied correctly:

• Addition: Combining numbers or terms.
• Subtraction: Finding the difference between numbers or terms.
• Multiplication: Repeated addition of a number or term.
• Division: Distributing a number or term into equal parts.

In our exercise, we focused on multiplication first: multiplying 1 by 21 and then by \(x\) to get \(21x\). Afterwards, subtraction was applied to simplify further by combining like terms.

Like Terms

Like terms are those that contain the same variable raised to the same power. They are vital for simplifying algebraic expressions because they allow us to combine terms and reduce the complexity of an expression.In the exercise, the expression \(16 - 21x - 4\) features constant like terms: 16 and -4. By combining these like terms, which do not include any variable part, we simplify the expression to \(12 - 21x\).If the expression had similar terms including variables, like \(21x\) and another \(x\) term, these would also be combined in the process.

Algebraic Expressions

An algebraic expression consists of numbers, variables, and arithmetic operators. It does not have an equal sign, as it's mainly used to represent mathematical relationships or patterns.Every component of an algebraic expression has its role:

• Constants: These are fixed numerical values.
• Variables: Symbols that represent unknown values.
• Coefficients: Numerical factors that multiply the variables.

In the expression \(16 - 1 \times 21 \times x - 4\), 16 and 4 are constants, \(x\) is the variable, and 21 is the coefficient of \(x\). By correctly applying the simplification steps, we turned a complex expression into a simpler form: \(12 - 21x\). This process is essential for effectively solving and understanding algebraic problems.