Problem 111 - Elementary and Intermediate Algebra

Question

IBM was founded 80 years before Apple Computer. Dell Computer Corporation was founded 9 years after Apple. a. Let \(x\) represent the age (in years) of one of the companies. Write expressions to represent the ages (in years) of the other two companies. b. On April \(1,2008\), Apple Computer Company was 32 years old. How old were the other two computer companies then?

Step-by-Step Solution

Answer

IBM is 112 years old, and Dell is 23 years old in 2008.

Step 1: Identify Known Ages and Variables

Let Apple's age be represented by the variable \( x \). Given that Apple was 32 years old in 2008, we have \( x = 32 \).

Step 2: Express IBM's Age

IBM was founded 80 years before Apple. Therefore, if Apple's age is \( x \), then IBM's age can be represented by \( x + 80 \). Substitute Apple's age, \( x = 32 \), into the expression: \( 32 + 80 = 112 \). Thus, IBM is 112 years old.

Step 3: Express Dell's Age

Dell was founded 9 years after Apple. Therefore, if Apple's age is \( x \), Dell's age can be represented by \( x - 9 \). Substitute Apple's age, \( x = 32 \), into the expression: \( 32 - 9 = 23 \). Thus, Dell is 23 years old.

Key Concepts

Understanding Expressions

In algebra, an expression is a combination of numbers, variables, and operations (like addition and subtraction) that represent a value. Expressions are used to describe relationships between different quantities. In the problem we are discussing, expressions help us calculate the ages of companies given certain relationships.

• An expression can be as simple as a single number, such as "5" or "10".
• It can include variables, like "x", representing unknown values we want to find.
• Operations are important in expressions, dictating how numbers and variables combine, like in "x + 80".

When solving age-related problems, creating expressions allows you to connect information given in the problem with what you want to find out. For example, figuring out how old IBM is involves the expression "x + 80", where "x" is Apple's age, and "80" is how many years older IBM is compared to Apple.

Introduction to Variables

Variables are symbols used to represent unknown values in mathematical expressions and equations. Often, we use letters like "x", "y", or "z" as variables, and they can take different numerical values depending on the problem.
In our age word problem, the variable "x" was chosen to represent Apple's age because it was initially unknown, or rather, it was a given that we later used numerically (here, "x = 32", because Apple was 32 years old in 2008).

• Variables allow for flexibility in representing different scenarios using mathematical models.
• They act as placeholders we can later substitute with specific numerical values as we solve parts of a problem.
• A key aspect of working with variables is understanding how they change within different expressions or equations to give meaningful answers.

In this scenario, using the variable "x" efficiently allowed us to easily calculate IBM's and Dell's ages through simple substitutions and arithmetic.

Algebraic Equations in Action

Algebraic equations are statements that, within an equality sign "=", express the equivalence between two mathematical expressions. An equation establishes a relationship that helps find the value of unknown variables. In the context of age problems like the one we’re solving, equations use known values to find unknowns.
For example, in the problem, the known equation is simply Apple's age, "x = 32", which becomes the cornerstone to solve for the ages of IBM and Dell.

• Equations include expressions that represent values on both sides of an "=" sign.
• The goal is to isolate the variable to solve for its value, turning the abstract into concrete.
• Solving comes down to simple arithmetic once the expressions are set, allowing you to determine exact values easily.

In this problem, you use "IBM's age = x + 80" and "Dell's age = x - 9" as equations, substituting "x = 32" to find exact ages for both IBM and Dell. Using equations in this structured way makes solving such word problems clearer and more straightforward.