Age Equations and Their Solutions: A Comparative Analysis

In today's lesson, we will delve into three distinct age-related algebraic problems that challenge our understanding of linear equations. We will explore the steps to solve these problems and verify the solutions to ensure their accuracy.

Problem 1: Tom's Age

Let's start with Tom's age. We are told that in two years, Tom will be twice as old as he was five years ago. To solve this, let's denote Tom's current age as x.

- Five years ago, Tom's age was x - 5.

- In two years, Tom will be x 2.

According to the problem, in two years Tom will be twice as old as he was five years ago:

x 2 2(x - 5)

Now we'll solve for x step by step.

x 2 2x - 10

2 x - 10 (Subtract x from both sides)

x 12 (Add 10 to both sides)

Therefore, Tom is 12 years old.

Problem 2: Rami's Age

Now, let's consider Rami's age. We are given that in two years, Rami will be twice as old as he was five years ago. Let's denote Rami's current age as x.

- Five years ago, Rami's age was x - 5.

- In two years, Rami will be x 2.

According to the problem, in two years Rami will be twice as old as he was five years ago:

x 2 2(x - 5)

Simplifying and solving for x step by step:

x 2 2x - 10

2 x - 10

x 12

Therefore, Rami is currently 12 years old.

Problem 3: Billy's Age

Next, we'll examine the age of Billy. We are told that in two years, Billy will be twice as old as he was five years ago. Let's denote Billy's current age as x.

- Five years ago, Billy's age was x - 5.

- In two years, Billy will be x 2.

According to the problem, in two years Billy will be twice as old as he was five years ago:

x 2 2(x - 5)

Simplifying the equation:

x 2 2x - 10

x 12

Therefore, Billy is currently 12 years old.

Conclusion

To ensure the accuracy of our solutions, we can substitute 12 into the original equations:

For Tom: T 2 2(T - 5) 14 2(7), which is true.

For Rami: R 2 2(R - 5) 14 2(7), which is true.

For Billy: B 2 2(B - 5) 14 2(7), which is true.

Thus, all the equations have been verified and confirmed to be correct.

These examples demonstrate the importance of systematically solving linear equations to find the correct age for a given problem. By following these steps, we can solve similar age-related algebraic problems.