Problem 128 - Algebra and Trigonometry

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. Eight decreased by three times the sum of a number and six.

Step-by-Step Solution

Answer

The simplified expression for the English phrase 'Eight decreased by three times the sum of a number and six.' is \(-3x - 10\).

Step 1: Interpret the English phrase as an algebraic expression

The key to translating word problems is understanding the language used. In this problem, 'Eight decreased by three times the sum of a number and six' translates into: \(8 - 3 \times (x + 6)\)

Step 2: Use the distributive property to simplify the expression

Apply the distributive property by multiplying 3 with each of the quantities inside the parenthesis: \(8 - 3x - 18\)

Step 3: Combine like terms

We have two constant terms (8 and -18) in our expression. Combine them to simplify the expression: \(-3x - 10\)

Key Concepts

Translating Word Problems Into Algebraic Expressions

Translating word problems into algebraic expressions requires understanding how language represents mathematical operations. Words like "sum," "product," "difference," and "quotient" are common indicators of addition, multiplication, subtraction, and division, respectively. In our example, the phrase "Eight decreased by three times the sum of a number and six" is a bit convoluted but can be broken down into clear mathematical terms.

- "Decreased by" tells you it's a subtraction operation where one quantity will be subtracted from another.- "Three times" indicates multiplication by three.- "The sum of a number and six" implies an addition operation within parentheses.

So, the phrase translates to:
"Eight decreased by three times (x + 6)".

Interpreting these clues accurately is crucial in forming the correct algebraic expression, in this case, \[8 - 3 \times (x + 6)\] which represents the problem mathematically.

The Distributive Property

The distributive property is a useful mathematical tool that allows you to multiply a single term across terms within parentheses. It's expressed as: \[a \times (b + c) = a \times b + a \times c\]This property is applied to simplify expressions that involve parentheses and multiplication.

In the example expression \[8 - 3 \times (x + 6)\],we use the distributive property to multiply 3 with both \(x\) and 6, within the parentheses. This is done by:

• Multiplying \(3\times x\) which gives \(-3x\).
• Multiplying \(3\times 6\) which gives \(-18\).

Thus, the expression becomes \[8 - 3x - 18\]. This simplification allows us to remove the parentheses and proceed to the next step of simplifying further by combining like terms.

Combining Like Terms

Combining like terms is the step where you simplify the expression by adding or subtracting terms that have the same variables and powers. In algebraic expressions, like terms have the same variables raised to the same power but can have different coefficients.

In \[8 - 3x - 18\], our task is to combine the constant terms, which are the numbers without variables. Here, those are 8 and -18. Combining them involves:
- Calculating \(8 - 18\) which simplifies to \(-10\).

Therefore, the expression now reads \[-3x - 10\]. Always look out for like terms in your expression to ensure you simplify it as much as possible, making it easier to work with or solve.