Problem 127 - College Algebra Essentials

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. The difference between eight times a number and six more than three times the number.

Step-by-Step Solution

Answer

The algebraic expression for the given English phrase is \(5x - 6\)

Step 1: Convert the phrase into an algebraic expression

The phrase 'eight times a number' can be represented as \(8x\). 'Six more than three times the number' refers to \(3x + 6\). The full phrase 'The difference between eight times a number and six more than three times the number.' is therefore represented as: \(8x - (3x + 6)\).

Step 2: Simplify the expression

Now simplify the expression using the order of operations (PEMDAS/BODMAS), where parentheses/brackets have the highest priority. Thus, first simplify the term in parentheses \(3x + 6\). Since there is no operation to perform, the parentheses can be removed, getting \(8x - 3x - 6\).

Step 3: Combine like terms

By combining the terms with \(x\), the simplified algebraic expression approximates to \(5x - 6\).

Key Concepts

Simplification

When dealing with algebraic expressions, simplification is a crucial step. Simplification involves reducing the expression to its simplest form while retaining its original value. In our problem, the expression begins as \(8x - (3x + 6)\). Here is where simplification comes into play.

• The parentheses around \(3x + 6\) need to be addressed first. Since there are no operations to compute inside them further, they can be eliminated.
• Once simplified to \(8x - 3x - 6\), the expression needs further reduction.

The goal of simplification is to make the expression easier to work with, facilitating any further calculations or analysis. Performing simplification correctly ensures that the essence of the expression remains, but in a more convenient form for additional computations.

Combining Like Terms

Combining like terms is one of the fundamental operations we use in simplifying expressions. Like terms are terms that have exactly the same variable raised to the same power. In the expression \(8x - 3x - 6\), the like terms are \(8x\) and \(3x\).Here's how we combine like terms:

• Identify like terms: In our case, \(8x\) and \(-3x\) both contain the variable \(x\).
• Subtract the coefficients of these like terms: \(8x - 3x\) results in \(5x\).
• The constant \(-6\) remains unaffected as it does not have a like term to combine with.

Finally, after combining, the expression transforms to \(5x - 6\). Combining like terms helps in reducing the expression, making it manageable, and bringing it toward its simple form.

Order of Operations

Understanding the order of operations is key to tackling algebraic expressions accurately. Often remembered by the acronym PEMDAS or BODMAS, it dictates the sequence in which mathematical operations should be carried out:

• P/B: Parentheses/Brackets - Solve expressions inside parentheses or brackets first.
• E/O: Exponents/Orders - Handle exponents or orders next.
• MD: Multiplication and Division - Perform these operations from left to right.
• AS: Addition and Subtraction - Finally, complete any addition or subtraction from left to right.

In our problem, the order of operations guided us to handle the parentheses in \(8x - (3x + 6)\) initially. As we progressed, eliminating these parentheses paved the way to simplify further by combining like terms. Keeping the order of operations in mind ensures that calculations are carried out in the right sequence, leading to correct solutions.