Problem 123 - Algebra and Trigonometry

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. Six times the product of negative five and a number.

Step-by-Step Solution

Answer

-30x

Step 1: Translation to mathematical term

The phrase translates to 'six multiplied by the result of negative five multiplied by a number'. Since the number is represented by x, the algebraic expression becomes 6*(-5*x)

Step 2: Simplify

Simplifying this expression involves multiplication. According to the rule of multiplying integers, the product of an even number of negative terms is positive, and the product of an odd number of negative terms is negative. Therefore, 6*(-5*x) simplifies to -30*x.

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra. It means rewriting an expression in its most compact and straightforward form without changing its value. Simplification makes it easier to understand and work with expressions.
Simplification often involves combining like terms, which are terms that have the same variable raised to the same power, or applying arithmetic operations like addition, subtraction, multiplication, and division.
For example, in the expression \(6 imes (-5 imes x)\), we can simplify by calculating the multiplication. The expression simplifies to \(-30x\), which is more straightforward.

Multiplication of Integers

Understanding the multiplication of integers is essential for simplifying algebraic expressions. It's important to know the rules for multiplying both positive and negative integers.
Here are some key rules:

• The product of two positive integers is positive.
• The product of two negative integers is positive.
• The product of a positive and a negative integer is negative.

In our example, \(6*(-5*x)\), we multiply 6 (a positive integer) by -5 and \(x\) (where \(-5 \times x = -5x\)). According to the multiplication rules, multiplying a positive by a negative results in a negative product.

Translation of Phrases to Expressions

This skill involves converting words into algebraic symbols and operations. It requires careful reading and understanding of phrases.
When translating, look for keywords:

• "Times" suggests multiplication.
• "Plus" or "sum" suggests addition.
• "Product" indicates multiplication as well.
• "Minus" or "less" means subtraction.

In the exercise, "six times the product of negative five and a number" translates to "6 * (-5 * x)." "Product" indicates multiplication, and "times" suggests another multiplication by six.

Negative Numbers in Multiplication

Multiplying with negative numbers can be confusing due to the "sign change" rules. It's crucial to apply these rules correctly to get the right result.
Remember:

• Multiplying two negative numbers gives a positive result because negatives cancel out each other.
• Multiplying a negative and a positive number yields a negative result because only one negative doesn't cancel out.

In our case, \(6*(-5*x)\), the multiplication of 6 by \(-5\) gives \(-30\), after which \(x\) is attached, resulting in \(-30x\). This demonstrates how the presence of one negative in the multiplication affects the entire expression's sign.