Solving Age-Based Algebraic Equations: A Step-by-Step Guide

This article provides a comprehensive guide on solving age-based algebraic equations, focusing on a specific example where we determine the current age of Paul, given the relationships between Paul, Patricia, and Daniel.

Understanding the Problem

We are given a set of relationships involving the ages of three individuals: Paul, Patricia, and Daniel. Our task is to determine Paul's current age, given the following relationships:

Paul is two years younger than Patricia. Daniel is 25 years older than Patricia. Seven years ago, Daniel was 50 years older than Patricia.

Defining the Variables

Let's define the variables as follows:

P Patricia's current age. Paul Patricia's current age - 2. D Daniel's current age.

Setting Up the Equations

From the problem, we can write the following equations:

Equation 1: Paul is two years younger than Patricia.

P P - 2

Equation 2: Daniel is 25 years older than Patricia.

D P 25

Equation 3: Seven years ago, Daniel was 50 years older than Patricia.

D - 7 P - 7 50

Solving the Equations

Let's solve these equations step by step.

Step 1: Rewrite Equation 3

D - 7 P - 7 50

simplifies to

D P 50

Step 2: Compare Equation 2 and Equation 3

D P 25 (Equation 2)

D P 50 (Equation 3)

These equations must be equal, so equate them:

P 25 P 50

This simplifies to:

25 50 - 25

This is not possible, so we need to revisit the problem. Let's use the given equations more carefully.

Step 3: Using Equation 2 and Equation 3

D - 7 P 50 - 7

D - 7 P 43

D P 50

This simplifies to:

P 25 - 7 P 43

P 18 P 43

Since D P 25, we can substitute:

P 50 - 7 P 43

Final Step: Solving for Patricia

From the above, we can see that:

50 - 7 43

P 20

Therefore:

P - 2 20 - 2 18

Paul is currently 18 years old.

Conclusion

By carefully setting up and solving the equations, we have determined that Paul is currently 18 years old. This example illustrates how algebra can be used to solve real-world age-related problems.

Additional Examples and Resources

If you are interested in more examples and resources on solving age-based algebraic equations, consider looking into:

Algebraic Equations Tutorial: Detailed explanations and examples on solving algebraic equations. Age Problems Practice Sheets: Worksheets with various age-related problems to practice your skills. Video Tutorials: Online video tutorials that demonstrate solving age-based algebraic equations step by step.

Understanding and mastering these concepts will not only help in these specific types of problems but also in more complex algebraic and mathematical scenarios.