Problem 69 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let \(x\) represent the number. The quotient of \(-20\) and a number, increased by the quotient of 3 and the number

Step-by-Step Solution

Answer

The English phrase translates to the algebraic expression \(-20/x + 3/x\) which simplifies to \(-17/x\).

Step 1: Translate English phrase to Algebraic expression: Part 1

The 'quotient of -20 and a number' can be translated to \(-20/x\). 'Quotient' refers to the result obtained from the division operation (-20 is being divided by the number \(x\)).

Step 2: Translate English phrase to Algebraic expression: Part 2

'Increased by the quotient of 3 and the number' can be translated to \(+ 3/x\) where 'increased by' signifies addition operation (adding the result of dividing 3 by the number \(x\)).

Step 3: Combine the expressions

Now both parts of the expression are combined. This is done by adding the two expressions together to get \(-20/x + 3/x\).

Step 4: Simplify the expression

Simplify the expression by combining the like terms (terms with the same variable part). The resulting simplified expression is \((-20 + 3)/x = -17/x\).

Key Concepts

Quotient

A **quotient** is the result of a division problem. It's what you get after dividing one number by another. In algebraic expressions, recognizing how to interpret the term "quotient" is crucial. The quotient is expressed with a fraction. For example, if we have "the quotient of 9 and a number," it converts into an algebraic formula as \( \frac{9}{x} \), where \( x \) is the variable representing the number we are dividing by. Here, division is the key element, and it helps us understand how different numbers relate to each other when they are divided.
To make sense of division involving a variable, it's essential to remember that the quotient shows how many times one number is contained within another. This concept is very useful when dealing with unknowns, as you often deal with variables in various forms.

Simplifying Expressions

**Simplifying expressions** makes them easier to work with. It involves combining like terms and reducing equations to their simplest form. For the expression \[ \frac{-20}{x} + \frac{3}{x} \]we notice both terms share the same denominator, \( x \). This allows us to directly combine the numerators through simple addition or subtraction. In this case, we combine \(-20\) and \(3\) to get \[ \frac{-17}{x} \]
When terms in fractions share the same denominator, they can be easily added or subtracted just like whole numbers. The result should be expressed as simply as possible, ensuring clarity and efficiency in both solving and understanding algebraic problems.

• Ensure all terms are similar before combining them.
• Simplification is key in making expressions less complex and more digestible.
• The process helps in error reduction and smoother calculations in further operations.

Variable Representation

**Variable representation** uses symbols, often letters like \( x \), to stand in for unknown values. This forms the basis of algebra. Variables allow you to create expressions and equations that describe relationships between numbers. For example, in our exercise, \( x \) is representing "a number." By doing this, we can write expressions general enough to work for any specific value of \( x \).
Choosing a variable means you can handle operations without knowing the specific number. It makes algebra versatile, as you can apply the same formula to multiple scenarios. Understanding how to assign and work with variables in expressions allows for dynamic modeling in mathematics.
Variables act as placeholders until specific numbers are substituted, and they are fundamental for creating general rules in arithmetic and algebra. By mastering variable representation, you can simplify complex problems and understand deeper concepts in mathematics.